Translation Functors and the Shapovalov Determinant

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 10 February 2015

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

The Virasoro Algebra

The Virasoro algebra is the Lie algebra $$\text{Vir}=ℂ\text{-span}\{z,d_k\hspace{0.17em}|\hspace{0.17em}k\in \mathbb{Z}\}$$ with bracket $[,]$ given by $$[d_k,z]=0,[d_j,d_k]=(j-k)d_{j+k}+\frac{{\delta }_{j,-k}}{12}(j^3-j)z\text{.}$$

([MPi1995], 1.9.4). The Virasoro algebra is the universal central extension of the Lie algebra $W=ℂ\text{-span}\left\{D_k\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}$ with relations $$[D_j,D_k]=(j-k)D_{j+k}\text{.}$$

See Section 1.3.5 for a definition of central extensions. The Lie algebra $W$ is the Witt algebra. The Witt algebra can be identified with the Lie algebra of derivations on $ℂ[t,t^{-1}]\text{.}$ That is, the derivations $\{D_k=-t^{k+1}\frac{d}{dt}\hspace{0.17em}|\hspace{0.17em}k\in \mathbb{Z}\}$ are a basis for the derivations of $ℂ[t,t^{-1}],$ and they satisfy the relation $D_j{D}_k-D_k{D}_j=(j-k)D_{j+k}\text{.}$

The Virasoro algebra has a regular, finite Hermitian triangular decomposition $\text{Vir}={\text{Vir}}_{-}\oplus 𝔥\oplus {\text{Vir}}_{+}$ where $$\begin{array}{rcl}{\displaystyle {\text{Vir}}_{-}}& =& {\displaystyle \text{span}\left\{d_n\hspace{0.17em}\right|\hspace{0.17em}n\in {\mathbb{Z}}_{<0}\}\text{;}}\\ {\displaystyle 𝔥}& =& {\displaystyle \text{span}\{d_0,z\}\text{;}}\\ {\displaystyle {\text{Vir}}_{+}}& =& {\displaystyle \text{span}\left\{d_n\hspace{0.17em}\right|\hspace{0.17em}n\in {\mathbb{Z}}_{>0}\}\text{.}}\end{array}$$ The associated Hermitian anti-involution on Vir is $\varphi :\text{Vir}\to \text{Vir}$ given by $$\varphi \left(d_n\right)=d_{-n},\varphi \left(z\right)=z\text{.}$$

Let $U\left(\text{Vir}\right)$ be the universal enveloping algebra of Vir. Proposition 2.0.5 shows that $U\left(\text{Vir}\right)$ inherits a triangular decomposition from Vir: $$\begin{array}{cc}{\displaystyle U\left(\text{Vir}\right)=U\left({\text{Vir}}_{-}\right)S\left(𝔥\right)U\left({\text{Vir}}_{+}\right),}& \text{(3.1)}\end{array}$$ where $S\left(𝔥\right)=U\left(𝔥\right)$ is the symmetric algebra of $𝔥\text{.}$ Bases for $U\left({\text{Vir}}_{-}\right)$ and $U\left({\text{Vir}}_{+}\right)$ are $\left\{d_{-{\lambda }_1}\cdots d_{-{\lambda }_k}\hspace{0.17em}\right|\hspace{0.17em}{\lambda }_i\in \mathbb{Z},{\lambda }_1\ge \cdots \ge {\lambda }_k>0\}$ and $\left\{d_{-{\lambda }_1}\cdots d_{-{\lambda }_k}\hspace{0.17em}\right|\hspace{0.17em}{\lambda }_i\in \mathbb{Z},{\lambda }_1\ge \cdots \ge {\lambda }_k>0\},$ respectively.

In order to write these bases more efficiently, we introduce the following notation. For a partition $\lambda :$ ${\lambda }_1\ge \cdots \ge {\lambda }_k>0,$ the weight of $\lambda$ is $\mid \lambda \mid ={\lambda }_1+\cdots +{\lambda }_k\text{.}$ The number of parts of $\lambda$ is denoted $l\left(\lambda \right)=k\text{.}$ Define $$\begin{array}{rcl}{\displaystyle d_{\lambda }}& =& {\displaystyle d_{{\lambda }_k}\cdots d_{{\lambda }_1}}\\ {\displaystyle d_{-\lambda }}& =& {\displaystyle d_{-{\lambda }_1}\cdots d_{-{\lambda }_k}\text{.}}\end{array}$$ Then, the bases for $U\left({\text{Vir}}_{-}\right)$ and $U\left({\text{Vir}}_{+}\right)$ can be rewritten as $\left\{d_{-\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \hspace{0.17em}\text{a partition}\}$ and $\left\{d_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \hspace{0.17em}\text{a partition}\},$ respectively.

Category $𝒪$

Category $𝒪$ was introduced in Section 2.1. We now make a few comments specific to the Virasoro algebra.

Recall ${𝔥}^{*}={\text{Hom}}_{ℂ}(𝔥,ℂ)\text{.}$ We can identify weights $\lambda \in {𝔥}^{*}$ with pairs in ${ℂ}^2$ by $$\lambda \leftrightarrow (\lambda \left(d_0\right),\lambda \left(z\right))\text{.}$$ Then, the partial ordering on ${𝔥}^{*}$ is given by $$\mu <\lambda \hspace{0.17em}\text{if}\hspace{0.17em}\mu \left(z\right)=\lambda \left(z\right)\hspace{0.17em}\text{and}\hspace{0.17em}\lambda \left(d_0\right)-\mu \left(d_0\right)\in {\mathbb{Z}}_{<0}\text{.}$$

For $\lambda \in {𝔥}^{*},$ the Verma module $M\left(\lambda \right)$ is the induced module $$M\left(\lambda \right)=U\left(\text{Vir}\right){\otimes }_{U({\text{Vir}}_{+}\oplus 𝔥)}{ℂ}_{\lambda }\text{.}$$ We write $M(h,c)$ for $M\left(\lambda \right),$ where $(h,c)$ is the pair identified with $\lambda \text{.}$ Similarly, we write $J(h,c)$ for the unique maximal submodule of $M(h,c)$ (Lemma 2.1.3) and $L(h,c)$ for the unique irreducible quotient of $M(h,c)\text{.}$

Using the PBW basis for $U\left({\text{Vir}}_{-}\right),$ we see $$M(h,c)=\underset{n\ge 0}{⨁}M{(h,c)}^{(h+n,c)},\text{where}\hspace{0.17em}\left\{d_{-\lambda }{v}^{+}\hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid =n\}\hspace{0.17em}\text{is a basis for}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\text{.}$$ Then $\text{dim}\hspace{0.17em}M{(h,c)}^{(h+n,c)}=p\left(n\right),$ where $p\left(n\right)$ is the number of partitions of $n\text{.}$ Recall from Section 2.1.1 that the character of a module $M$ records the dimensions of the weight spaces of $M\text{.}$ For $M(h,c),$ we have $$\begin{array}{rcll}{\displaystyle \text{ch}\left(M(h,c)\right)}& =& {\displaystyle \sum _{n=1}^{\infty }p\left(n\right)e^{(h+n,c)}}& \text{(3.2)}\\ {\displaystyle }& =& {\displaystyle \frac{e^{(h,c)}}{\prod _{j=1}^{\infty }(1-q^j)}}& \text{(3.3)}\end{array}$$ where $g=e^{(1,0)}\text{.}$

Affine Lie Algebras

The Virasoro algebra has a close relationship with affine Lie algebras. In particular, it is possible to construct a representation of the Virasoro algebra on certain modules for affine Lie algebras. Before discussing this construction, we provide a brief introduction to affine Lie algebras.

Recall from Section 1.2.1 that a reductive Lie algebra $𝔤$ is a direct sum of an abelian Lie algebra and simple Lie algebras. Let $𝔤$ be a finite-dimensional reductive Lie algebra with nondegenerate bilinear form $(,)\text{.}$ (We assume this is the Killing form when $𝔤$ is semisimple.) The affine Lie algebra $\widehat{𝔤}$ associated to $𝔤$ is $$\widehat{𝔤}=(ℂ[t,t^{-1}]\otimes 𝔤)\oplus ℂc\oplus ℂd,$$ with relations $$\begin{array}{c}[t^m\otimes x,t^n\otimes y]=t^{m+n}\otimes [x,y]+m{\delta }_{m,-n}(x,y)c,\\ [c,t^m\otimes x]=0,[c,d]=0,[d,t^n\otimes x]=n{t}^n\otimes x,\end{array}$$ for $m,n\in \mathbb{Z}$ and $x,y\in 𝔤\text{.}$ We define $$\widehat{𝔤}\prime =[\widehat{𝔤},\widehat{𝔤}]=ℂ[t,t^{-1}]\otimes 𝔤+ℂc\text{.}$$ Finally, observe that $𝔤\subseteq \widehat{𝔤}$ via the identification $x\mapsto 1\otimes x\text{.}$ For $m\in \mathbb{Z}$ and $x\in 𝔤,$ we adopt the notation $$x\left(m\right)=t^m\otimes x\text{.}$$

We extend the form $(,)$ to a bilinear form on all of $\widehat{𝔤}$ by $$\begin{array}{c}(x\left(n\right),y\left(m\right))={\delta }_{n,-m}(x,y),(x\left(n\right),c)=0,(x\left(n\right),d)=0,\\ (c,d)=1,(c,d)=0,(d,d)=0\text{.}\end{array}$$ Then $(,)$ is nondegenerate on $\widehat{𝔤}\text{.}$

The amne Lie algebra $\widehat{𝔤}$ has a triangular decomposition with Cartan subalgebra $$\widehat{𝔥}=𝔥\oplus ℂd\oplus ℂc.$$ Then ${\widehat{𝔥}}^{*}$ has a basis $\{{\alpha }_1,\dots ,{\alpha }_n,\delta ,\zeta \}$ where $\left\{{\alpha }_i\right\}$ are the simple roots of $𝔤$ and $$\begin{array}{c}\delta \left(𝔥\right)=0\delta \left(d\right)=1\delta \left(c\right)=0\text{;}\\ \zeta \left(𝔥\right)=0\zeta \left(d\right)=0\zeta \left(c\right)=1\text{.}\end{array}$$ The bilinear form $⟨,⟩$ on ${𝔥}^{*}$ extends to all of ${\widehat{𝔥}}^{*}$ by $$\begin{array}{rcl}{\displaystyle ⟨\delta ,{\alpha }_i⟩}& =& {\displaystyle 0=⟨\zeta ,{\alpha }_i⟩,}\\ {\displaystyle ⟨\delta ,\zeta ⟩}& =& {\displaystyle 1}\\ {\displaystyle ⟨\delta ,\delta ⟩}& =& {\displaystyle 0=⟨\zeta ,\zeta ⟩\text{.}}\end{array}$$

Restricted Modules and the Casimir Element

A $\widehat{𝔤}$ module $V$ is restricted if for all $v\in V,$ $x\left(n\right)v=0$ for each $x\in 𝔤$ for $n$ sufficiently large. In particular, simple modules $L\left(\lambda \right)$ (since they are highest weight modules) are restricted modules. The restricted completion $\widehat{U\left(\stackrel{ˆ}{𝔤}\right)}$ of $U\left(\widehat{𝔤}\right)$ is the set of infinite sums $\sum _{i=1}^{\infty }{x}_i,$ $x_i\in U\left(\widehat{𝔤}\right),$ such that for any restricted module $V$ and $v\in V,$ $x_{i}v=0$ for all but finitely many $x_i\text{.}$ Two sums are considered the same if they act the same on all restricted modules. (See [Kac1104219], 2.5 and 12.8 for more on these definitions.)

Let $𝔤$ be a simple Lie algebra. Let $\left\{u_i\hspace{0.17em}\right|\hspace{0.17em}1\le i\le \text{dim}\hspace{0.17em}𝔤\}$ be a basis for $𝔤,$ and let $u^i$ be a dual basis with respect to $(,),$ so that $(u_i,u^j)={\delta }_{i,j}\text{.}$ We define the Casimir element for $\widehat{𝔤}$ to be $$\mathrm{\Omega }=2(c+g)d+\sum _i{u}^i{u}_i+2\sum _{n=1}^{\infty }\sum _i{u}^i(-n)u_i\left(n\right)$$ where $g=\frac{1}{2}⟨\theta +2\rho ,\theta ⟩$ is the dual Coxeter number of $𝔤\text{.}$ (Recall $\rho =\frac{1}{2}\sum _{\beta \in R_{+}}\beta \text{.)}$ Observe that $\mathrm{\Omega }\in \widehat{U\left(\stackrel{ˆ}{𝔤}\right)}\text{.}$

([Kac1104219], Theorem 2.6 and Corollary 2.6).

(i)	Let $x\in \widehat{𝔤}\text{.}$ Then as operators on a restricted $\widehat{𝔤}\text{-module,}$ $[\mathrm{\Omega },x]=0\text{.}$
(ii)	For $\lambda \in {𝔥}^{*},$ $\mathrm{\Omega }$ acts on $L\left(\lambda \right)$ by $⟨\lambda +2\hat{\rho },\lambda ⟩,$ where $\hat{\rho }=\rho +\zeta \text{.}$

The Virasoro Algebra and Affine Lie Algebras

We now construct an action of the Virasoro algebra on restricted $\widehat{𝔤}\text{-modules.}$ This construction, known as the Sugawara construction, follows [KRa1987].

Let $𝔤$ be a simple or abelian Lie algebra and let $\widehat{𝔤}$ be the associated affine Lie algebra. Recall that the Virasoro algebra is the universal central extension of the Lie algebra of differential operators on $ℂ[t,t^{-1}]\text{.}$ Let $$D_k=t^{k+1}\frac{d}{dt}\text{.}$$ These operators have a natural action on $\widehat{𝔤}\prime$ given by $$\begin{array}{rcl}{\displaystyle [D_k,x\left(n\right)]}& =& {\displaystyle t^{k+1}\frac{d}{dt}(t^n\otimes x)=nx(n+k)}\\ {\displaystyle [D_k,c]}& =& {\displaystyle 0\text{.}}\end{array}$$ Note that $[D_0,x\left(n\right)]=nx\left(n\right)=[d,x\left(n\right)]\text{;}$ that is, the action of $D_0$ on $\widehat{𝔤}\prime$ coincides with the action of $d\text{.}$ We would like to define an action of Vir on $\widehat{𝔤}\text{-modules}$ that is consistent with these relations, For $k\in \mathbb{Z}$ let $$T_k=\frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _{1\le i\le \text{dim}\hspace{0.17em}𝔤}:u_i(-n)u^i(n+k):\in \widehat{U\left(\stackrel{ˆ}{𝔤}\right)},$$ where the normal ordering $:·:$ is $$:x(-n)y\left(m\right):≔\{\begin{array}{ll}x(-n)y\left(m\right)& \text{if}\hspace{0.17em}-n\le m,\\ y\left(m\right)x(-n)& \text{if}\hspace{0.17em}-n>m\text{.}\end{array}$$ Note that $$\begin{array}{cc}{\displaystyle T_0=\frac{1}{2}\mathrm{\Omega }-2(c+g)d\text{.}}& \text{(3.4)}\end{array}$$ As we will show, the operators $T_k\in \widehat{U\left(\stackrel{ˆ}{𝔤}\right)}$ mimic the action of $D_k$ on $\widehat{𝔤}\prime \text{.}$

For all $k,m\in \mathbb{Z}$ and $x\in 𝔤,$ $[T_k,x\left(m\right)]=-(c+g)mx(m+k)\text{.}$

		Proof.
	Let $m,n\in \mathbb{Z}\text{.}$ From Equations 1.3 and 1.4, and Proposition 1.2.3 we have $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \sum _i[u_i\left(m\right),u^i\left(n\right)]}& =& {\displaystyle \sum _i[u_i,u^i](m+n)+\kappa (u_i,u^i)m{\delta }_{m,-n}c}\\ & =& {\displaystyle m{\delta }_{m,-n}c\hspace{0.17em}\text{dim}\hspace{0.17em}𝔤\text{;}}\end{array}}& \text{(3.5)}\end{array}$$ $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \sum _i[[x,u_i]\left(m\right),u^i\left(n\right)]}& =& {\displaystyle \sum _i[[x,u_i],u^i](m+n)}\\ & =& {\displaystyle 2gx(m+n)\text{;}}\end{array}}& \text{(3.6)}\end{array}$$ and $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \sum _i[x,u_i]\left(m\right)u^i\left(n\right)}& =& {\displaystyle \sum _{i,j}\kappa ([x,u_i],u^j)u_j\left(m\right)u^i\left(n\right)}\\ & =& {\displaystyle \sum _{i,j}-\kappa (u_i,[x,u^j])u_j\left(m\right)u^i\left(n\right)}\\ & =& {\displaystyle \sum _{i,j}-u_j\left(m\right)\kappa ([x,u^j],u_i)u^i\left(n\right)}\\ & =& {\displaystyle -\sum _j{u}_j\left(m\right)[x,u^j]\left(n\right)\text{.}}\end{array}}& \text{(3.7)}\end{array}$$ We then have $$\begin{array}{rcl}{\displaystyle [x\left(m\right),T_k]}& =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i\underset{\underset{\underset{\text{that we can ignore the normal ordering.}}{\text{Since}\hspace{0.17em}[x\left(m\right),c]=0\text{, Equation 3.5 implies}}}{⏟}}{[x\left(m\right),u_i(-n)u^i(n+k)]}}\\ & =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i[x\left(m\right),u_i(-n)]u^i(n+k)+u_i(-n)[x\left(m\right),u^i(n+k)]}\\ & =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i[x,u_i](m-n)u^i(n+k)+m{\delta }_{m,n}\kappa (x,u_i)c{u}^i(n+k)}\\ & & {\displaystyle +\frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i{u}_i(-n)[x,u^i](m+n+k)+m{\delta }_{m,-(n+k)}\kappa (x,u^i)c{u}_i(-n)}\\ & =& {\displaystyle \underset{\underset{*}{⏟}}{\frac{1}{2}\sum _{n<\frac{m-k}{2}}\sum _i[x,u_i](m-n)u^i(n+k)}+\underset{\underset{**}{⏟}}{\frac{1}{2}\sum _{n\ge \frac{m-k}{2}}\sum _i[x,u_i](m-n)u^i(n+k)}}\\ & & {\displaystyle +\underset{\underset{♣}{⏟}}{\frac{1}{2}\sum _{n<\frac{m+k}{2}}\sum _i{u}_i(-n)[x,u^i](m+n+k)}}\\ & & {\displaystyle +\underset{\underset{♣♣}{⏟}}{\frac{1}{2}\sum _{n\ge \frac{m+k}{2}}\sum _i{u}_i(-n)[x,u^i](m+n+k)}}\\ & & {\displaystyle +mx(m+k)c}\\ & =& {\displaystyle \underset{\underset{\text{We reorder}\hspace{0.17em}*\hspace{0.17em}\text{using Equation 3.6 and combine with}\hspace{0.17em}**\text{.}}{⏟}}{\frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i:[x,u_i](m-n)u^i(n+k):+\sum _{n<\frac{m-k}{2}}gx(m+k)}}\\ & & {\displaystyle +\underset{\underset{\text{We reorder}\hspace{0.17em}♣\hspace{0.17em}\text{using Equation 3.6 and combine with}\hspace{0.17em}♣♣\text{.}}{⏟}}{\sum _{n\in \mathbb{Z}}\sum _i:u_i(-n)[x,u^i](m+n+k):-\sum _{n<-\frac{m+k}{2}}gx(m+k)}}\\ & & {\displaystyle +mx(m+k)c}\\ & =& {\displaystyle \underset{\underset{\underset{\text{line. Equation 3.7 implies this sum is}\hspace{0.17em}0\text{.}}{\text{We let}\hspace{0.17em}n\to n+m\hspace{0.17em}\text{in the first sum from the previous}}}{⏟}}{\sum _{n\in \mathbb{Z}}\sum _i:[x,u_i](-n)u^i(m+n+k)+u_i(-n)[x,u^i](m+n+k):}}\\ & & {\displaystyle +mx(m+k)c+\sum _{-\frac{m+k}{2}\le n<\frac{m-k}{2}}gx(m+k)}\\ & =& {\displaystyle mx(m+k)(c+g)\text{.}}\end{array}$$ $\square$

For $j,k\in \mathbb{Z},$ $[T_j,T_k]=(c+g)(j-k)T_{j+k}+{\delta }_{j,-k}\frac{j^3-j}{12}\left(\text{dim}\hspace{0.17em}𝔤\right)c(c+g)\text{.}$

		Proof.
	$$\begin{array}{rcl}{\displaystyle [T_j,T_k]}& =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i[T_j,u_i(-n)u^i(n+k)]}\\ & =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i\underset{\underset{\underset{\text{ordering from Equation 3.5}}{\text{We can again ignore the normal}}}{⏟}}{T_j,u_i(-n)u^i(n+k)+u_i(-n)[T_j,u^i(n+k)]}}\\ & =& {\displaystyle \frac{1}{2}\sum _{n\in \mathbb{Z}}\sum _i\underset{\underset{\text{From the previous lemma}}{⏟}}{n{u}_i(-n+j)u^i(n+k)(c+g)-(n+k)u_i(-n)u^i(n+j+k)(c+g)}}\\ & =& {\displaystyle \frac{1}{2}(c+g)\sum _{n\in \mathbb{Z}}\sum _{i}n:u_i(-n+j)u^i(n+k):}\\ & & {\displaystyle +\frac{1}{2}(c+g)\sum _{n<\frac{j-k}{2}}-n(-n+j){\delta }_{j,-k}\text{dim}\left(𝔤\right)c}\\ & & {\displaystyle +\frac{1}{2}(c+g)\sum _{n\in \mathbb{Z}}\sum _i-(n+k):u_i(-n)u^i(n+j+k):}\\ & & {\displaystyle +\frac{1}{2}(c+g)\sum _{n<-\frac{j+k}{2}}n(n+k){\delta }_{j,-k}\text{dim}\left(𝔤\right)c}\\ & =& {\displaystyle \frac{1}{2}(c+g)\sum _{n\in \mathbb{Z}}\sum _i(j-k):u_i(-n)u^i(n+j+k):}\\ & & {\displaystyle +\frac{1}{2}c(c+g)\sum _{0\le n<j}n(-n+j)\text{dim}\left(𝔤\right)c}\\ & =& {\displaystyle (j-k)c(c+g)T_{j+k}+\frac{j^3-j}{12}\text{dim}\left(𝔤\right)c(c+g)}\end{array}$$ $\square$

Suppose that $V$ is a $\widehat{𝔤}\text{-module}$ where $c$ acts by a scalar $M\text{.}$ We will call $M$ the level of $V\text{.}$

Suppose that $V$ is a restricted $\widehat{𝔤}\text{-module}$ with level $M\ne -g\text{.}$ Then $$d_k\mapsto \frac{1}{M+g}{T}_k$$ defines an action of $\text{Vir}$ on $V$ with $$z\mapsto \frac{M}{M+g}\hspace{0.17em}\text{dim}\hspace{0.17em}𝔤\text{.}$$

		Proof.
	This follows from the previous lemma. $\square$

Let $𝔤$ be a reductive Lie algebra. Then $𝔤=\underset{i=1}{\overset{k}{⨁}}{𝔤}_i,$ where ${𝔤}_i$ is simple or abelian. For each $1\le i\le k,$ suppose $V_i$ is a restricted ${\widehat{𝔤}}_i^{\prime }\text{-module,}$ with level $M_i\text{.}$ The above construction gives an action of Vir on $V_i\text{;}$ denote the action of $d_k$ by $d_k^{{𝔤}_i}\text{.}$

Note that $\widehat{𝔤}\prime =\underset{i=1}{\overset{k}{⨁}}{\widehat{𝔤}}_i^{\prime }\text{.}$ Therefore, the tensor product $V_1\otimes \cdots \otimes V_k$ is a $\widehat{𝔤}\prime \text{-module}$ (where ${\widehat{𝔤}}_i^{\prime }$ acts on $V_i\text{).}$ Define $$d_k^{𝔤}=\sum _{i=1}^k{d}_k^{{𝔤}_i}\text{.}$$

The map $d_k\mapsto d_k^{𝔤}$ defines a representation of $\text{Vir}$ on $V$ with $$z\mapsto \sum _{i=1}^k{z}^{{𝔤}_i}=\sum _{i=1}^k\frac{M_i}{M_i+g_i}\hspace{0.17em}\text{dim}\hspace{0.17em}{𝔤}_i\text{.}$$

		Proof.
	Since ${\widehat{𝔤}}_i^{\prime }$ commutes with ${\widehat{𝔤}}_j^{\prime },$ the operators $d_k^{{𝔤}_i}$ and $d_k^{{𝔤}_j}$ commute, and the result follows. $\square$

For the proof of Theorem 3.4.3, we will use a slight modification of the above construction. Let $𝔤$ be a reductive Lie algebra and let $𝔭\subseteq 𝔤$ be a reductive subalgebra of $𝔤\text{.}$ Then, for any restricted $𝔤\text{-module,}$ we can construct representations of Vir corresponding to both $𝔤$ and $𝔭\text{.}$ Denote the Vir-operators corresponding to $𝔤$ and $𝔭$ by $d_k^{𝔤},$ $z^{𝔤}$ and $d_k^{𝔭},$ $z^{𝔭},$ respectively. Define an action of $d_k$ on restricted $𝔤\text{-modules}$ by $$d_k\mapsto d_k^{𝔤-𝔭}=d_k^{𝔤}-d_k^{𝔭}\text{.}$$

The operators $d_k$ form a representation of the Virasoro algebra with $z$ acting by $z\mapsto z^{𝔤-𝔭}=z^{𝔤}-z^{𝔭}\text{.}$ Moreover, the action of $\text{Vir}$ commutes with the action of $\widehat{𝔭}\prime \text{.}$

		Proof.
	Since $[d_k^{𝔤},x\left(n\right)]=nx(n+k)=[d_k^{𝔭},t^{n}x\left(n\right)]$ for $x\left(n\right)\in \widehat{𝔭}\prime ,$ $[d_k^{𝔤-𝔭},\widehat{𝔭}\prime ]=0\text{.}$ This implies $[d_k^{𝔤-𝔭},d_k^{𝔭}]=0\text{.}$ Therefore $$\begin{array}{rcl}{\displaystyle [d_j^{𝔤-𝔭},d_k^{𝔤-𝔭}]}& =& {\displaystyle [d_j^{𝔤},d_k^{𝔤}]-[d_j^{𝔭},d_k^{𝔭}]}\\ & =& {\displaystyle (j-k)d_{j+k}^{𝔤-𝔭}+{\delta }_{j,-k}\frac{j^3-j}{12}(z^{𝔤}-z^{𝔭})\text{.}}\end{array}$$ $\square$

The Affine Lie Algebra $\widehat{{sl}_2\left(ℂ\right)}$

The proof of the determinant formula given in the next section relies specifically on the representions of the Virasoro algebra on $\widehat{{sl}_2\left(ℂ\right)}\text{-modules.}$

We fix the simple root $\alpha$ of ${sl}_2\left(ℂ\right)\text{.}$ Then, ${\widehat{𝔥}}^{*}=ℂ\alpha \oplus ℂ\delta \oplus ℂ\zeta \text{.}$

([KRa1987] 11.4 and 12.1). Let $\lambda \in {\widehat{𝔥}}^{*}$ such that $\lambda =m\zeta +\frac{n}{2}\alpha ,$ $m\ge n\ge 0\text{.}$ Set $q=e^{-\zeta }\text{.}$ Then, $$\text{ch}\left(L\left(\lambda \right)\right)\text{ch}\left(L\left(\zeta \right)\right)=\sum _{k\in I}{\psi }_{m,n,k}\text{ch}\left(L(\zeta +\lambda -k\alpha )\right)$$ where $$\begin{array}{rcl}{\displaystyle I}& =& {\displaystyle \{k\in \mathbb{Z}\hspace{0.17em}|\hspace{0.17em}-\frac{1}{2}(m+1-n)\le k\le \frac{n}{2}\}\text{;}}\\ {\displaystyle {\psi }_{m,n,k}}& =& {\displaystyle \frac{(f_{m,n,k}-f_{m,n,n+1-k})}{\prod _{j=1}^{\infty }(1-q^j)}\text{;}}\\ {\displaystyle f_{m,n,k}}& =& {\displaystyle \sum _{j\in \mathbb{Z}}{q}^{(m+2)(m+3)j^2+(n+1+2k(m+2))j+k^2}\text{.}}\end{array}$$ Also, $$\begin{array}{cc}{\displaystyle \text{ch}\left(L\left(\lambda \right)\right)\text{ch}\left(L\left(\zeta \right)\right)=\sum _{k\in I}\sum _{j\in {\mathbb{Z}}_{\ge 0}}{\mathrm{\Delta }}_{m,n,k}^j\text{ch}\left(L(\zeta +\lambda -k\alpha -j\delta )\right)}& \text{(3.8)}\end{array}$$ where ${\mathrm{\Delta }}_{m,n,k}^j\in {\mathbb{Z}}_{\ge 0}$ are such that ${\psi }_{m,n,k}=\sum _{j\in {\mathbb{Z}}_{\ge 0}}{\mathrm{\Delta }}_{m,n,k}^j{q}^j\text{.}$ The minimum value of $j$ for which ${\mathrm{\Delta }}_{m,n,k}^j$ is nonzero is $k^2\text{.}$

For dominant integral weights $\mu$ and $\gamma$ of $\widehat{{sl}_2\left(ℂ\right)},$ the tensor product of $L\left(\mu \right)\otimes L\left(\gamma \right)$ is completely reducible ([Kac1104219], Corollary 10.7). (A weight $\mu \in {\widehat{𝔥}}^{*}$ is dominant integral if $⟨\mu ,\alpha ⟩,⟨\mu ,\delta ⟩,⟨\mu ,\zeta ⟩\in {\mathbb{Z}}_{\ge 0}\text{.)}$ Therefore, Equation 3.8 implies that as $\widehat{{sl}_2\left(ℂ\right)}\text{-modules}$ $$\begin{array}{rcl}{\displaystyle L\left(\zeta \right)\otimes L\left(\lambda \right)}& \cong & {\displaystyle \underset{\underset{j\in {\mathbb{Z}}_{\ge 0}}{k\in I}}{⨁}L{(\zeta +\lambda -k\alpha -j\delta )}^{\oplus {\mathrm{\Delta }}_{m,n,k}^j}}\\ & \cong & {\displaystyle \underset{\underset{j\ge k^2}{k\in I}}{⨁}L{(\zeta +\lambda -k\alpha -j\delta )}^{\oplus {\mathrm{\Delta }}_{m,n,k}^j}\text{.}}\end{array}$$

We use the construction from the previous section, with $𝔭={sl}_2\left(ℂ\right)$ and $𝔤={sl}_2\left(ℂ\right)\oplus {sl}_2\left(ℂ\right)\text{.}$ (We embed ${sl}_2\left(ℂ\right)$ in ${sl}_2\left(ℂ\right)\oplus {sl}_2\left(ℂ\right)$ via the diagonal map: $x\mapsto x\oplus x\text{.)}$

Let $V$ and $W$ be restricted $\widehat{{sl}_2\left(ℂ\right)}\text{-modules.}$ For $x\oplus y\in \widehat{{sl}_2\left(ℂ\right)}\oplus \widehat{{sl}_2\left(ℂ\right)}$ and $u\otimes w\in V\otimes W,$ define $(x\oplus y)(v\otimes w)=xv\otimes w+v\otimes yw\text{.}$ Therefore, $$\begin{array}{cc}{\displaystyle d_k^{𝔤}=d_k^{{sl}_2\left(ℂ\right)}\otimes 1+1\otimes d_k^{{sl}_2\left(ℂ\right)}\text{.}}& \text{(3.10)}\end{array}$$

Let $\lambda \in {\widehat{𝔥}}^{*}$ such that $\lambda =m\zeta +\frac{n}{2}\alpha ,$ $m\ge n\ge 0\text{.}$ We consider the action of Vir on $L\left(\lambda \right)\otimes L\left(\zeta \right)\text{.}$ Proposition 3.2.1 and Equation 3.4 imply that

•	$d_0^{{sl}_2\left(ℂ\right)}$ acts on $L\left(\lambda \right)$ by $\frac{(\lambda +2\hat{\rho },\lambda )}{m+2}-2d\text{;}$
•	$d_0^{{sl}_2\left(ℂ\right)}$ acts on $L\left(\zeta \right)$ by $\frac{(\zeta +2\hat{\rho },\zeta )}{1+2}-2d\text{;}$
•	$d_0^{𝔭}$ acts on $L\left(\lambda \right)\otimes L\left(\zeta \right)$ by $\frac{1}{m+1+2}\mathrm{\Omega }-2(d\otimes 1+1\otimes d)\text{.}$

(The dual Coxeter number of ${sl}_2\left(ℂ\right)$ is $g=2\text{.)}$ Therefore, $d_0^{𝔤-𝔭}$ acts on $L\left(\lambda \right)\otimes L\left(\zeta \right)$ by $$\begin{array}{rcll}{\displaystyle d_0^{𝔤-𝔭}}& =& {\displaystyle d_0^{𝔤}-d_0^{𝔭}=d_0^{{sl}_2\left(ℂ\right)}\otimes 1+1\otimes d_0^{{sl}_2\left(ℂ\right)}-d_0^{𝔭}}& \text{(3.11)}\\ & =& {\displaystyle \frac{1}{2}(\frac{⟨\lambda +2\hat{\rho },\lambda ⟩}{m+2}+\frac{⟨\zeta +2\hat{\rho },\zeta ⟩}{1+2}-\frac{1}{m+1+2}\mathrm{\Omega })}\\ {\displaystyle }& =& {\displaystyle \frac{n(n+2)}{4(m+2)}-\frac{1}{2(m+3)}\mathrm{\Omega }}& \text{(3.12)}\end{array}$$ Also, $$\begin{array}{cc}{\displaystyle z^{𝔤-𝔭}=\frac{m}{m+2}3+\frac{1}{1+2}3-\frac{m+1}{m+1+2}3=1-\frac{6}{(m+2)(m+3)}\text{.}}& \text{(3.13)}\end{array}$$

A Determinant Formula for $M\left(\lambda \right)$

Recall the Hermitian anti-involution $\varphi :U\left(\text{Vir}\right)\to U\left(\text{Vir}\right)$ denned by $\varphi \left(d_k\right)=d_{-k},$ $\varphi \left(z\right)=z\text{.}$ For $(h,v)\in {ℝ}^2,$ we use this to define an Hermitian form $⟨,⟩:M(h,c)\times M(h,c)\to ℂ$ by

•	$⟨v^{+},v^{+}⟩=1,$ where $v^{+}$ is a (fixed) generator of $M(h,c)\text{;}$
•	$⟨xv,\stackrel{\sim }{v}⟩=⟨v,\varphi \left(x\right)\stackrel{\sim }{v}⟩$ for $x\in U\left(\text{Vir}\right),$ $v,\stackrel{\sim }{v}\in M(h,c)\text{.}$

As we saw in the previous chapter, the form $⟨,⟩$ has two important properties:

•	$M{(h,c)}^{(h+n,c)}\perp M{(h,c)}^{(h+m,c)}$ for $m\ne n$ (Lemma 2.4.1);
•	$\text{Rad}⟨,⟩=J(h,c)$ (Lemma 2.4.2).

Therefore, the determinant $\text{det}\left(M{(h,c)}^{(h+n,c)}\right)=\text{det}{(d_{-\lambda }{v}^{+},d_{-\stackrel{\sim }{\lambda }}{v}^{+})}_{\mid \lambda \mid =n=\mid \stackrel{\sim }{\lambda }\mid }$ provides a tool to study $J(h,c)\text{.}$

Example. Below, $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ (for $(h,c)\in {ℝ}^2\text{)}$ is computed for $n=1,2\text{.}$ $$\begin{array}{rcl}{\displaystyle \text{det}\hspace{0.17em}M{(h,c)}^{(h+1,c)}}& =& {\displaystyle \text{det}\left(⟨d_{-1}{v}^{+},d_{-1}{v}^{+}⟩\right)}\\ & =& {\displaystyle ⟨v^{+},d_1{d}_{-1}{v}^{+}⟩}\\ & =& {\displaystyle ⟨v^{+},(d_{-1}{d}_1+2d_0)v^{+}⟩}\\ & =& {\displaystyle ⟨v^{+},2h{v}^{+}⟩}\\ & =& {\displaystyle 2h\text{.}}\end{array}$$ $$\begin{array}{rcl}{\displaystyle \text{det}\hspace{0.17em}M{(h,c)}^{(h+2,c)}}& =& {\displaystyle \text{det}\left(\begin{array}{cc}⟨d_{-1}^2{v}^{+},d_{-1}^2{v}^{+}⟩& ⟨d_{-2}{v}^{+},d_{-1}^2{v}^{+}⟩\\ ⟨d_{-1}^2{v}^{+},d_{-2}{v}^{+}⟩& ⟨d_{-2}{v}^{+},d_{-2}{v}^{+}⟩\end{array}\right)}\\ & =& {\displaystyle \text{det}\left(\begin{array}{cc}8h^2+4h& 6h\\ 6h& 4h+c/2\end{array}\right)}\\ & =& {\displaystyle 2h(16h^2+2(c-5)h+c)}\end{array}$$Theorem 3.4.3 gives a general formula for $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\text{.}$

The highest power of $h$ in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ is $$\sum _{\underset{1\le rs\le n}{r,s\in {\mathbb{Z}}_{>0}}}p(n-rs),$$ and the coefficient of this term is $$\prod _{\underset{1\le r\le s\le n}{r,s\in {\mathbb{Z}}_{>0}}}{({\left(2r\right)}^{s}s!)}^{p(n-rs)-p(n-r(s+1))}\text{.}$$

		Proof.
	Consider the entries of $A{(h,c)}^{(h+n,c)}={\left(⟨d_{-\lambda }{v}^{+},d_{-\stackrel{\sim }{\lambda }}⟩\right)}_{\mid \lambda \mid =n=\mid \stackrel{\sim }{\lambda }\mid }\text{.}$ Let $\lambda$ and $\stackrel{\sim }{\lambda }$ be partitions of $n\text{.}$ Writing $d_{\lambda }{d}_{-\stackrel{\sim }{\lambda }}$ in terms of the decomposition of $U\left(\text{Vir}\right)$ in Equation 3.1, we have that $$\begin{array}{cc}{\displaystyle d_{\lambda }{d}_{-\stackrel{\sim }{\lambda }}=\sum _{\nu ,\mu \hspace{0.17em}\text{partitions}}{d}_{-\nu }{p}_{\nu ,\mu }(d_0,z)d_{-\mu },}& \text{(3.14)}\end{array}$$ where $p_{\nu ,\mu }(d_0,z)$ is a polynomial in $d_0$ and $z\text{.}$ Then $⟨d_{-\lambda }{v}^{+},d_{-\stackrel{\sim }{\lambda }}{v}^{+}⟩=⟨v^{+},d_{\lambda }{d}_{-\stackrel{\sim }{\lambda }}{v}^{+}⟩=p_{0,0}(h,c)\text{.}$ Now consider $p_{0,0}(h,c)$ more closely. We can use the relations $$\begin{array}{rcl}{\displaystyle d_j{d}_k}& =& {\displaystyle d_k{d}_j+(j-k)d_{j+k}\text{if}\hspace{0.17em}j\ne -k}\\ {\displaystyle d_k{d}_{-k}}& =& {\displaystyle 2k{d}_0+\frac{k^3-k}{12}z}\end{array}$$ to rearrange $d_{\lambda }{d}_{-\stackrel{\sim }{\lambda }}$ as in (3.14). These imply that, as a polynomial in $h,$ the degree of $\left(p_{0,0}(h,c)\right)$ is less than or equal to $l\left(\lambda \right),l\left(\stackrel{\sim }{\lambda }\right)\text{;}$ and the degree of $\left(p_{0,0}(h,c)\right)=l\left(\lambda \right)$ if and only if $\lambda =\stackrel{\sim }{\lambda }\text{.}$ Therefore, for any given row of $A{(h,c)}^{(h+n,c)},$ the entry with the highest powers of $h$ is the diagonal entry. Thus, the highest power of $h$ in the determinant comes from the product of the diagonal entries in $A{(h,c)}^{(h+n,c)},$ The degree of this term is $$\begin{array}{rcl}{\displaystyle \sum _{\mid \lambda \mid =n}l\left(\lambda \right)}& =& {\displaystyle \sum _{\underset{1\le rs\le n}{r,s\in {\mathbb{Z}}_{>0}}}\underset{\underset{\text{A partition with}\hspace{0.17em}t\hspace{0.17em}\text{parts of size}\hspace{0.17em}r\hspace{0.17em}\text{will be counted for}\hspace{0.17em}s=1,2,\dots ,t\text{.}}{⏟}}{\text{the number of partitions of}\hspace{0.17em}n\hspace{0.17em}\text{with at least}\hspace{0.17em}s\hspace{0.17em}\text{parts of size}\hspace{0.17em}r}}\\ & =& {\displaystyle \sum _{\underset{1\le rs\le n}{r,s\in {\mathbb{Z}}_{>0}}}\underset{\underset{\underset{\text{we obtain a partition of}\hspace{0.17em}n-rs\text{.}}{\text{By removing}\hspace{0.17em}s\hspace{0.17em}\text{parts of size}\hspace{0.17em}r}}{⏟}}{p(n-rs)}}\end{array}$$ We now compute the coefficient of this term. A partition may be written as $\lambda =(r_1^{s_1},\dots ,r_j^{s_j})\text{.}$ Note that $$\begin{array}{rcl}{\displaystyle d_r{d}_{-r}^s}& =& {\displaystyle (d_{-r}{d}_r+2r{d}_0+\frac{r^3-r}{12}z)d_{-r}^{s-1}}\\ & =& {\displaystyle d_{-r}{d}_r{d}_{-r}^{s-1}+d_{-r}^{s-1}(2r{d}_0+2r^2(s-1)+\frac{r^3-r}{12}z)}\\ & =& {\displaystyle d_{-r}^s{d}_r+d_{-r}^{s-1}(2rs{d}_0+r^{2}s(s-1)+\frac{(r^3-r)s}{12}z),}\end{array}$$ and so $d_r^s{d}_{-r}^s=d_{-r}^s{d}_r^s+{\left(2r\right)}^{s}s!d_0^s+$ terms of lower degree in $d_0\text{.}$ Therefore, the coefficient of the highest power of $h$ in a diagonal entry $⟨d_{-r_1}^{s_1}\cdots d_{-r_j}^{s_j}{v}^{+},d_{-r_1}^{s_1}\cdots d_{-r_j}^{s_j}{v}^{+}⟩$ is ${\left(2r_j\right)}^{s_j}\left(s_j\right)!,$ and the coefficient of the highest power of $h$ in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ is $$\prod _{\underset{\lambda =(r_1^{s_1},\dots ,r_j^{s_j})}{\mid \lambda \mid =n}}{\left(2r_j\right)}^{s_j}\left(s_j\right)!=\prod _{\underset{1\le r\le s\le n}{r,s\in {\mathbb{Z}}_{>0},}}{({\left(2r\right)}^{s}s!)}^{p(n-rs)-p(n-r(s+1))}\text{.}$$ $\square$

Since the highest power of $h$ in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ does not involve $c,$ we fix $c$ and think of $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ as a polynomial in $h\text{.}$

Fix $c\in ℝ\text{.}$ Let $h_0\in ℝ$ and suppose $(h-h_0)$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+k,c)}\text{.}$ Then ${(h-h_0)}^{p(n-k)}$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\text{.}$

		Proof.
	Suppose $(h-h_0)$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+k,c)}\text{.}$ This implies $A_k(h_0,c)$ is degenerate. In other words, there is a vector ${(a_1,\dots ,a_{p\left(k\right)})}^T,$ $a_i\in ℂ$ and $a_i\ne 0$ for at least one $i,$ such that $A_k(h_0,c){(a_1,\dots ,a_{p\left(k\right)})}^T=0\text{.}$ Then. $$A_k(h,c)\left(\begin{array}{c}{a}_1\\ ⋮\\ a_{p\left(k\right)}\end{array}\right)=\left(\begin{array}{c}{P}_1\\ ⋮\\ P_{p\left(k\right)}\end{array}\right),$$ where the $P_i$ are polynomials in $h$ which are divisible by $(h-h_0)\text{.}$ Define $\stackrel{\sim }{v}=\sum _{i=1}^{p\left(k\right)}{a}_i{d}_{-{\lambda }^{\left(i\right)}}{v}^{+}\text{.}$ We then have $(h-h_0)$ divides $P_i=⟨d_{-{\lambda }^{\left(i\right)}}{v}^{+},\stackrel{\sim }{v}⟩\text{.}$ Consider $\stackrel{\sim }{B}=\left\{d_{-\lambda }\left(\sum _{i=1}^{p\left(k\right)}{a}_i{d}_{-{\lambda }^{\left(i\right)}}\right)\hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid =n-k\}\subseteq U{\left({\text{Vir}}_{-}\right)}^{(n,0)}\text{.}$ (Here we view $U\left({\text{Vir}}_{-}\right)$ as a Vir-module under the adjoint action.) This set is linearly independent in $U\left({\text{Vir}}_{-}\right)$ and can be extended to a set $B$ of basis vectors for $U{\left({\text{Vir}}_{-}\right)}^{(n,0)}\text{.}$ Let $P$ be the matrix taking $B$ to $\left\{d_{-\lambda }\hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid =n\}\text{.}$ Then the entries of $P$ are in $ℂ$ and $\text{det}\left(P\right)\ne 0\text{.}$ Now, for $d_{-\lambda }\left(\sum _{i=1}^{p\left(k\right)}{a}_i{d}_{-{\lambda }^{\left(i\right)}}\right)\in \stackrel{\sim }{B},$ $$d_{-\lambda }\sum _{i=1}^{p\left(k\right)}{a}_i{d}_{-{\lambda }^{\left(i\right)}}{v}^{+}=d_{-\lambda }\stackrel{\sim }{v}\text{.}$$ Also, $(h-h_0)$ divides $⟨d_{-\lambda }\stackrel{\sim }{v},w⟩$ for all $w\in M(h,c)\text{.}$ Then $${(h-h_0)}^{p(n-k)}|\text{det}{\left(⟨X_i{v}^{+},X_j{v}^{+}⟩\right)}_{X_i,X_j\in \bar{B}}\text{.}$$ Finally, $$\begin{array}{rcl}{\displaystyle \text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}}& =& {\displaystyle \text{det}\left(P^t{\left(⟨X_i{v}^{+},X_j{v}^{+}⟩\right)}_{X_i,X_j\in \bar{B}}P\right)}\\ & =& {\displaystyle \text{det}\left(P^t\right)\text{det}\left(P\right)\text{det}{\left(⟨X_i{v}^{+},X_j{v}^{+}⟩\right)}_{X_i,X_j\in \bar{B}}}\\ & =& {\displaystyle \text{det}{\left(P\right)}^2\text{det}{\left(⟨X_i{v}^{+},X_j{v}^{+}⟩\right)}_{X_i,X_j\in \bar{B}}\text{.}}\end{array}$$ Since $\text{det}\left(P\right)\ne 0,$ this implies ${(h-h_0)}^{p(n-k)}$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\text{.}$ $\square$

([KRa1987], [FFu1990]). For $(h,c)\in {ℝ}^2$ and $n\in {\mathbb{Z}}_{\ge 0},$ $$\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}=\prod _{\underset{1\le r\le s\le n}{r,s\in {\mathbb{Z}}_{>0},}}{({\left(2r\right)}^{s}s!)}^{p(n-rs)-p(n-r(s+1))}\prod _{r,s\in {\mathbb{Z}}_{>0}}{(h-h_{r,s})}^{p(n-rs)},$$ where $$h_{r,s}\left(c\right)=\frac{1}{48}((13-c)(r^2+s^2)+\sqrt{(c-1)(c-25)}(r^2-s^2)-24rs-2+2c)\text{.}$$

		Proof.
	Given lemmas 3.4.1 and 3.4.2, we only need to show that $(h-h_{r,s}\left(c\right))$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+rs,c)}\text{.}$ We will use the representation of Vir on restricted $\widehat{{sl}_2\left(ℂ\right)}\text{-modules}$ to prove this. Recall (from Equation 3.9) that, for $\lambda =m\zeta +\frac{n}{2}\alpha$ $\text{(}m\ge n>0\text{),}$ we can write the tensor product $L\left(\zeta \right)\otimes L\left(\lambda \right)$ of $\widehat{{sl}_2\left(ℂ\right)}\text{-modules}$ as $$\begin{array}{rcl}{\displaystyle L\left(\zeta \right)\otimes L\left(\lambda \right)}& =& {\displaystyle \underset{\underset{j\in {\mathbb{Z}}_{\ge 0}}{k\in I}}{⨁}L{(\zeta +\lambda -k\alpha -j\delta )}^{\oplus {\mathrm{\Delta }}_{m,n,k}^j},}\\ & =& {\displaystyle \underset{\underset{j\ge k^2}{k\in I}}{⨁}L{(\zeta +\lambda -k\alpha -j\delta )}^{\oplus {\mathrm{\Delta }}_{m,n,k}^j}\text{.}}\end{array}$$ Let $U_{m,n,k}^j$ be the space of highest weight vectors of weight $\zeta +\lambda -k\alpha -j\delta$ in $L\left(\zeta \right)\otimes L\left(\lambda \right)\text{.}$ Then, $\text{dim}\hspace{0.17em}{U}_{m,n,k}^j={\mathrm{\Delta }}_{m,n,k}^j$ and $U_{m,n,k}=\underset{j\in {\mathbb{Z}}_{\ge 0}}{⨁}{U}_{m,n,k}^j$ is the space of highest weight vectors of weight $\zeta +\lambda -k\alpha$ for $\widehat{{sl}_2\left(ℂ\right)}\prime \text{.}$ From Section 3.3, we know that $L\left(\lambda \right)\otimes L\left(\zeta \right)$ is a Vir-module. Since the action of $\widehat{{sl}_2\left(ℂ\right)}\prime$ and Vir commute (Proposition 3.3.5), we also have that $U_{m,n,k}$ is a Vir-module. Moreover, given Equations 3.11 and 3.13 and Proposition 3.2.1, it is clear that $U_{m,n,k}^j$ is a weight space for the action of Vir such that • for all $v\in U_{m,n,k},$ $zv=(1-\frac{6}{(m+2)(m+3)})v\text{;}$ • for $v\in U_{m,n,k}^j,$ $$\begin{array}{rcl}{\displaystyle d_{0}v}& =& {\displaystyle (\frac{n(n+2)}{4(m+2)}-\frac{1}{2(m+3)}\mathrm{\Omega })v}\\ & =& {\displaystyle (\frac{n(n+2)}{4(m+2)}+j-\frac{(n-2k)(n-2k+2)}{4(m+3)})v\text{.}}\end{array}$$ Define $$\begin{array}{rcl}{\displaystyle h_{r,s}^m}& =& {\displaystyle \frac{{((m+3)r-(m+2)s)}^2-1}{4(m+2)(m+3)}}\\ {\displaystyle c_m}& =& {\displaystyle 1-\frac{6}{(m+2)(m+3)},}\end{array}$$ where $$\begin{array}{rcl}{\displaystyle r}& =& {\displaystyle n+1,}\\ {\displaystyle r}& =& {\displaystyle m-n+1,}\end{array}\begin{array}{rcll}{\displaystyle s}& =& {\displaystyle n+1-2k}& \text{if}\hspace{0.17em}k\ge 0\\ {\displaystyle s}& =& {\displaystyle m-n+2+2k,}& \text{if}\hspace{0.17em}k<0\text{.}\end{array}$$ Note that $(r,s)\mapsto (m+2-r,m+3-s)$ switches these definitions. $\square$

According to Proposition 3.3.6, the minimum value of $j$ for which $U_{m,n,k}^j\ne 0$ is $j=k^2\text{.}$ Therefore, as a Vir-module, $U_{m,n,k}$ has highest weight $(h_{r,s}^m,c_m)\text{.}$ Since ${\mathrm{\Delta }}_{m,n,k}^j<\infty ,$ this shows that $U_{m,n,k}\in {𝒪}_{\text{Vir}}\text{.}$ The character of $U_{m,n,k}$ is $$\begin{array}{rcl}{\displaystyle \text{ch}\hspace{0.17em}{U}_{m,n,k}}& =& {\displaystyle e^{(h_{r,s}^m,c_m)}\sum _{j\ge k^2}\text{dim}\hspace{0.17em}{U}_{m,n,k}^j{q}^{j-k^2}}\\ & =& {\displaystyle e^{(h_{r,s}^m,c_m)}{q}^{-k^2}{\psi }_{m,n,k}}\\ & =& {\displaystyle e^{(h_{r,s}^m,c_m)}\frac{q^{-k^2}}{\prod _{i=1}^{\infty }(1-q^i)}(f_{m,n,k}-f_{m,n,n+1-k})}\\ & =& {\displaystyle e^{(h_{r,s}^m,c_m)}\frac{1}{\prod _{i=1}^{\infty }(1-q^i)}}\\ & & {\displaystyle \times (\sum _{j\in \mathbb{Z}}{q}^{(m+2)(m+3)j^2+((m+3)r-(m+2)s)}-\sum _{j\in \mathbb{Z}}{q}^{(m+2)(m+3)j^2+((m+3)r+(m+2)s)+rs})}\\ & =& {\displaystyle \underset{\underset{=\text{ch}\hspace{0.17em}M(h_{r,s}^m,c_m)}{⏟}}{e^{(h_{r,s}^m,c_m)}\frac{1}{\prod _{i=1}^{\infty }(1-q^i)}}(1-q^{rs}-q^{(m+2-r)(m+3-s)}+\text{terms of degree}>\stackrel{\sim }{r}\stackrel{\sim }{s})\text{.}}\end{array}$$ (Here we are intentionally confusing $q=e^{(1,0)}$ (for the Virasoro algebra) and $q=e^{-\zeta }$ (for $\widehat{{sl}_2\left(ℂ\right)}\text{).)}$ Since the maximum weight for $U_{m,n,k}$ is $(h_{r,s}^m,c_m),$ $L(h_{r,s}^m,c_m)\subseteq U_{m,n,k}\text{.}$ Therefore, $\text{ch}\hspace{0.17em}L(c_m,h_{r,s}^m)\le \text{ch}\hspace{0.17em}{U}_{m,n,k}^m\text{.}$ Let $(\stackrel{\sim }{r},\stackrel{\sim }{s})$ be whichever of the pairs $(r,s),$ $(m+2-r,m+3-s)$ has minimum product. Note that $h_{\stackrel{\sim }{r},\stackrel{\sim }{s}}^m=h_{r,s}^m\text{.}$ The coefficient of $q^{\stackrel{\sim }{r}\stackrel{\sim }{s}}$ in $\text{ch}\hspace{0.17em}L(h_{r,s}^m,c_m)$ is less than the coefficient of $q^{\stackrel{\sim }{r}\stackrel{\sim }{s}}$ in $\text{ch}\hspace{0.17em}M(h_{r,s}^m,c_m),$ implying $\text{dim}\hspace{0.17em}L{(h_{r,s}^m,c_m)}^{(h_{r,s}^m+\stackrel{\sim }{r}\stackrel{\sim }{s},c_m)}<\text{dim}\hspace{0.17em}M{(h_{r,s}^m,c_m)}^{(h_{r,s}^m+\stackrel{\sim }{r}\stackrel{\sim }{s},c_m)}\text{.}$ Then, $J{(h_{r,s}^m,c_m)}^{(h_{r,s}^m+\stackrel{\sim }{r}\stackrel{\sim }{s},c_m)}=\text{Rad}\hspace{0.17em}{⟨,⟩}^{(h_{r,s}^m+\stackrel{\sim }{r}\stackrel{\sim }{s},c_m)}\ne 0\text{.}$ Since $\stackrel{\sim }{r}\stackrel{\sim }{s}\le rs,$ $J{(h_{r,s}^m,c_m)}^{(h_{r,s}^m+rs,c_m)}\ne 0\text{.}$ We then have $\text{det}\left(M{(h_{r,s}^m,c_m)}^{(h_{r,s}^m+rs,c_m)}\right)=0\text{.}$ Since $h_{r,s}\left(c_m\right)=h_{r,s}^m,$ $\text{det}\hspace{0.17em}M{(h,c)}^{(h+rs,c)}$ vanishes at infinitely many points along the curve $h=h_{r,s}\left(c\right)\text{.}$ Therefore, $(h-h_{r,s}\left(c\right))$ divides $\text{det}\hspace{0.17em}M{(h,c)}^{(h+rs,c)}\text{.}$

Blocks

Recall that we define an equivalence relation $\sim$ on the weights ${𝔥}^{*}$ of Vir generated by the relation $\lambda \sim \mu$ if $[M\left(\lambda \right):L\left(\mu \right)]>0\text{.}$ The blocks of the Virasoro algebra are the equivalence classes of $\sim \text{.}$ Prom Theorem 2.3.6, we know that $[M\left(\lambda \right):L\left(\mu \right)]>0$ if and only if $M\left(\mu \right)\subseteq M\left(\lambda \right)\text{.}$ We will use this alternative formulation of $\sim$ in order to describe the blocks of Vir.

Blocks and the Determinant Formula

For $r,s\in \mathbb{Z},$ define $${𝒞}_{r,s}(h,c)=\{\begin{array}{ll}{(h-\frac{1}{48}((13-c)(r^2+s^2)-24rs-2+2c))}^2& r\ne s\\ -\frac{1}{{48}^2}(c-1)(c-25){(r^2-s^2)}^2& \\ h-\frac{(r^2-1)(1-c)}{24}& r=s\end{array}$$ Viewing ${𝒞}_{r,s}(h,c)$ as a polynomial in $h,$ we have $${𝒞}_{r,s}(h,c)=\{\begin{array}{ll}(h-h_{r,s}\left(c\right))(h-h_{s,r}\left(c\right))& r\ne s,\\ h-h_{r,r}\left(c\right)& r=s\text{.}\end{array}$$ Therefore, the determinant formula for $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}$ can be rewritten as $$\begin{array}{cc}{\displaystyle \text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}=\prod _{\underset{1\le r\le s\le n}{r,s\in {\mathbb{Z}}_{>0},}}{({\left(2r\right)}^{s}s!)}^{p(n-rs)-p(n-r(s+1))}{\left({𝒞}_{r,s}(h,c)\right)}^{p(n-rs)}\text{.}}& \text{(3.15)}\end{array}$$ For $(h,c)\in {ℝ}^2,$ we know that $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}=0$ if and only if $J{(h,c)}^{(h+n,c)}\ne 0\text{.}$ Equation 3.15 implies that if $r,s\in {\mathbb{Z}}_{>0}$ are such that the product $rs$ is minimal with ${𝒞}_{r,s}(h,c)=0,$ then $J{(h,c)}^{(h+rs,c)}\ne 0$ and $J{(h,c)}^{(h+n,c)}=0$ for all $n<rs\text{.}$ Therefore, any vector $0\ne v\in J{(h,c)}^{(h+rs,c)}$ is a highest weight vector and so $M(h+rs,c)\subseteq M(h,c)\text{.}$ Theorem 3.5.1 shows that for any $r,s\in {\mathbb{Z}}_{>0}$ such that ${𝒞}_{r,s}(h,c)=0,$ $M(h+rs,c)\subseteq M(h,c)$ and that these embeddings produce a complete description of the submodule structure of $M(h,c)\text{.}$

For fixed $r,s,$ $r\ne s,$ the curves ${𝒞}_{r,s}(h,c)=0$ are hyperbolas. Below is the curve ${𝒞}_{1,2}(h,c)=0\text{.}$ $$\begin{array}{c} -20 c 20 40 -4 -2 h 4 2 \end{array}$$ Also note that

•	${𝒞}_{r,s}(h,c)={𝒞}_{-r,-s}(h,c),$
•	${𝒞}_{-r,s}(h,c)=0$ if and only if ${𝒞}_{r,s}(h-rs,c)=0,$
•	${𝒞}_{r,s}(h,c)={𝒞}_{-r,s}(1-h,26-c)\text{.}$

For fixed $(h,c)\in {ℝ}^2,$ ${𝒞}_{r,s}(h,c)$ can be factored into terms linear in $r$ and $s\text{:}$ $${𝒞}_{r,s}(h,c)=K(pr+qs+m)(pr+qs-m)(qr+ps+m)(qr+ps-m),$$ where $K,p,q,m\in ℂ$ such that $$\frac{p}{q}+\frac{q}{p}=\frac{c-13}{6}\text{;}4pqh+{(p+q)}^2=m^2\text{;}K=\frac{16}{p^2{q}^2}\text{.}$$ Thus, for fixed $(h,c),$ the solutions to the equation ${𝒞}_{r,s}(h,c)=0$ form two sets of parallel lines. The figure below illustrates the example ${𝒞}_{r,s}(0,0)=0\text{.}$ $$\begin{array}{c} -1.0 -0.5 0.5 1.0 -2 -1 1 2 \end{array}$$

To find all integer solutions to ${𝒞}_{r,s}(h,c)=0,$ we only need to consider one line, say $pr+qs+m=0\text{.}$ (If $(r,s)$ is a point on any of the other lines, $(-r,-s),$ $(s,r)$ or $(-s,-r)$ will lie on the line $pr+qs+m=0\text{.)}$ We fix one of the lines and call it ${ℒ}_{(h,c)}\text{.}$ Theorem 3.5.1 will show that the integer points $(r,s)$ on this line encode the embeddings $M(h\prime ,c\prime )\subseteq M(h,c)\subseteq M(h″,c″)\text{.}$

Note that a line passes through 0, 1, or infinitely many integer points. (If the line passes through two integer points, it has rational slope and therefore passes through infinitely many integer points.) In other words, there are 0, 1, or infinitely many curves ${𝒞}_{r,s}(h,c)=0,$ $r,s\in \mathbb{Z},$ passing through a fixed point $(h,c)\text{.}$ Below we include a partial picture of curves ${𝒞}_{r,s}(h,c)=0$ for values of $c$ near $c=1\text{.}$ $$\begin{array}{c} 0.2 0.4 0.6 0.8 1.0 c 1 2 3 4 5 h \end{array}$$ There are three points in the picture where multiple curves intersect: $(h,c)=(0,1),(1,1),(4,1)\text{.}$ As Theorem 3.5.1 shows, these weights belong to the block $\left[(0,1)\right]=\left\{(m^2,1)\hspace{0.17em}\right|\hspace{0.17em}m\in \mathbb{Z}\}\text{.}$

The line $ℒ(h,c)$ has nonzero slope. Thus, if it passes through infinitely many integer points $(r,s)$ with $rs>0$ it must pass through finitely many points $(r,s)$ with $rs<0,$ and vice versa.

([FFu1990]). Suppose $r,s\in {\mathbb{Z}}_{>0}$ such that ${𝒞}_{r,s}(h,c)=0\text{.}$ Then, $M(h+rs,c)\subseteq M(h,c)\text{.}$ All embeddings of Verma modules arise in this way. Therefore, we have the following description of Verma module embeddings.
$$Fix a pair $(h,c)\in {ℝ}^2,$ and let ${ℒ}_{(h,c)}$ be one of the lines defined by this pair. Then the Verma module embeddings involving $M(h,c)$ are described by one of the following four cases.

(i)	Suppose ${ℒ}_{(h,c)}$ passes through no integer points. The Verma module $M(h,c)$ is irreducible and does not embed in any other Verma modules. The block $\left[(h,c)\right]$ is given by $\left[(h,c)\right]=\left\{(h,c)\right\}\text{.}$
(ii)	Suppose ${ℒ}_{(h,c)}$ passes through exactly one integer point $(r,s)\text{.}$ (a) If $rs>0,$ the embeddings for $M(h,c)$ look like $$\begin{array}{cl}•& M(h,c)\\ \uparrow & \\ •& M(h+rs,c)\end{array}$$ where the arrow indicates inclusion. (b) If $rs<0,$ the embeddings for $M(h,c)$ look like $$\begin{array}{cl}•& M(h+rs,c)\\ \uparrow & \\ •& M(h,c)\end{array}$$ The block $\left[(h,c)\right]$ is given by $\left[(h,c)\right]=\{(h,c),(h+rs,c)\}\text{.}$
(iii)	Suppose ${ℒ}_{(h,c)}$ passes through infinitely many integer points and crosses an axis at an integer point. Label these points $(r_i,s_i)$ so that $\dots <r_{-2}{s}_{-2}<r_{-1}{s}_{-1}<0<r_1{s}_1<r_2{s}_2\dots \text{.}$ (We exclude points $(r,s)$ where $r=0$ or $s=0\text{;}$ these correspond to the embedding $M(h,c)=M(h+0,c)\subseteq M(h,c)\text{.}\text{)}$ $$\begin{array}{c} (r-1,s-1) (r1,s1) (r2,s2) (r3,s3) (r4,s4) (r5,s5) (r6,s6) (r7,s7) (r8,s8) (r9,s9) \end{array}$$ $$The embeddings between the corresponding Verma modules take one of the following forms: $$\begin{array}{cl}•& \\ \uparrow & \\ •& \\ ⋮& \\ •& M(h+r_{-1}{s}_{-1},c)\\ \uparrow & \\ •& M(h,c)\\ \uparrow & \\ •& M(h+r_1{s}_1,c)\\ ⋮& \\ \multicolumn{2}{c}{\text{slope}\hspace{0.17em}\left({ℒ}_{(h,c)}\right)>0}\end{array}\begin{array}{cl}⋮& \\ •& M(h+r_{-1}{s}_{-1},c)\\ \uparrow & \\ •& M(h,c)\\ \uparrow & \\ •& M(h+r_1{s}_1,c)\\ ⋮& \\ •& \\ \uparrow & \\ •& \\ \multicolumn{2}{c}{\text{slope}\hspace{0.17em}\left({ℒ}_{(h,c)}\right)<0}\end{array}$$ The block $\left[(h,c)\right]$ is given by $\left[(h,c)\right]=\{(h,c),(h+r_i{s}_i,c)\}\text{.}$
(iv)	Suppose ${ℒ}_{(h,c)}$ passes through infinitely many integer points and does not cross either axis at an integer point. Again label the integer points $(r_i,s_i)$ on ${ℒ}_{(h,c)}$ so that $\dots <r_{-2}{s}_{-2}<r_{-1}{s}_{-1}<0<r_1{s}_1<r_2{s}_2\dots \text{.}$ Also consider the auxiliary line ${\stackrel{\sim }{ℒ}}_{(h,c)}$ with the same slope as ${ℒ}_{(h,c)}$ passing through the point $(-r_1,s_1)\text{.}$ Label the integer points on this line $({\stackrel{\sim }{r}}_j,{\stackrel{\sim }{s}}_j)$ as above. The embeddings between the corresponding Verma modules take one of the forms $$\begin{array}{c} slope(ℒ(h,c))>0 slope(ℒ(h,c))<0 M(h+r1s1+r∼-3s∼-3,c) M(h+r1s1+r∼-4s∼-4,c) M(h+r-1s-1,c) M(h+r-2s-2,c) M(h+r1s1+r∼-1s∼-1,c) M(h,c)=M(h+r1s1+r∼-2s∼-2,c) M(h+r1s1,c) M(h+r2s2,c) M(h+r1s1+r∼1s∼1,c) M(h+r1s1+r∼2s∼2,c) M(h+r1s1+r∼-3s∼-3,c) M(h+r1s1+r∼-4s∼-4,c) M(h+r-1s-1,c) M(h+r-2s-2,c) M(h+r1s1+r∼-1s∼-1,c) M(h,c)=M(h+r1s1+r∼-2s∼-2,c) M(h+r1s1,c) M(h+r2s2,c) M(h+r1s1+r∼1s∼1,c) M(h+r1s1+r∼2s∼2,c) \end{array}$$ The block $\left[(h,c)\right]$ is given by $\left[(h,c)\right]=\{(h+r_i{s}_i,c),(h+r_1{s}_1+{\stackrel{\sim }{r}}_j{\stackrel{\sim }{s}}_j,c)\}\text{.}$

See Section 2.5 for more on Jantzen filtrations.

([FFu1990]). Let $(h,c)\in {ℝ}^2$ and classify $(h,c)$ according to the cases given above. Then the Jantzen filtration of $M(h,c)$ is given as follows:

(i), (iia)	$M{(h,c)}_j=0$ for all $j>0\text{.}$
(iib)	$M{(h,c)}_1=M(h+rs,c)$ and $M{(h,c)}_j=0$ for all $j>1\text{.}$
(iii)	$M{(h,c)}_j=M(h+r_j{s}_j,c)$ and $M{(h,c)}_j=0$ if there is no point $(r_j,s_j)$ on the line ${ℒ}_{(h,c)}\text{.}$ We have the following picture of the Jantzen filtration of $M(h,c)\text{:}$ $$\begin{array}{c} M(h,c) M(h,c)1 M(h,c)2 \end{array}$$
(iv)	Write $$\begin{array}{rclrcl}{\displaystyle n_{1,1}}& =& {\displaystyle r_1{s}_1}& {\displaystyle n_{1,2}}& =& {\displaystyle r_2{s}_2}\\ {\displaystyle n_{2,1}}& =& {\displaystyle r_1{s}_1+{\stackrel{\sim }{r}}_1{\stackrel{\sim }{s}}_1}& {\displaystyle n_{2,2}}& =& {\displaystyle r_1{s}_1+{\stackrel{\sim }{r}}_2{\stackrel{\sim }{s}}_2}\\ {\displaystyle n_{3,1}}& =& {\displaystyle r_3{s}_3}& {\displaystyle n_{3,2}}& =& {\displaystyle r_4{s}_4}\\ ⋮& & & ⋮\end{array}$$ Then $M{(h,c)}_j=M(h+n_{j,1},c)+M(h+n_{j,2},c),$ and $M(h+n_{j,1},c)\cap M(h+n_{j,2},c)=M{(h,c)}_{j+1}\text{.}$ We have the following picture of the Jantzen filtration of $M(h,c)\text{:}$ $$\begin{array}{c} M(h,c) M(h,c)1 M(h,c)2 M(h,c)3 \end{array}$$

		Partial Proof of Theorems 3.5.1 and 3.5.2.
	We give a proof of cases (i) and (ii) for both theorems simultaneously. We note that if we set $\gamma =(1,0),$ then for any $\lambda \in {𝔥}^{*},$ ${⟨,⟩}_{\lambda +t\gamma }$ will be nondegenerate. Therefore, Theorem 2.5.1 holds. Case (i): Suppose ${ℒ}_{(h,c)}$ passes through no integer points. Then, $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\ne 0$ for all $n\in {\mathbb{Z}}_{\ge 0}$ and so $M(h,c)$ is irreducible. Case (ii)a: Suppose ${ℒ}_{(h,c)}$ passes through one integer point $(r,s)$ with $rs>0\text{.}$ Since ${𝒞}_{r,s}(h,c)={𝒞}_{-r,-s}(h,c),$ we can assume that $r,s\in {\mathbb{Z}}_{>0}$ and that $r$ and $s$ are the only positive integers such ${𝒞}_{r,s}(h,c)=0\text{.}$ Therefore, $M(h+rs,c)\subseteq M(h,c)\text{.}$ This means $M(h+rs,c)\subseteq J(h,c),$ and so $$\text{dim}\hspace{0.17em}J{(h,c)}^{(h+n,c)}\ge \text{dim}\hspace{0.17em}M{(h+rs,c)}^{(h+n,c)}=p(n-rs)$$ for all $x\in {\mathbb{Z}}_{\ge 0}\text{.}$ However, from Theorem 2.5.1, $$\text{dim}\hspace{0.17em}J{(h,c)}^{(h+n,c)}\le \sum _{j\in {\mathbb{Z}}_{>0}}\text{dim}\hspace{0.17em}M{(h,c)}_j^{(h+n,c)}=\text{ord}\left(\text{det}\hspace{0.17em}M{(h+t,c)}^{(h+t+n,c)}\right)=p(n-rs)\text{.}$$ Therefore $M(h+rs,c)=J(h,c)=M{(h,c)}_1$ and $M{(h,c)}_j=0$ for $j>1\text{.}$ Since ${𝒞}_{r,s}(h+rs,c)={𝒞}_{-r,s}(h,c),$ there are no integers $r,s\in {\mathbb{Z}}_{>0}$ such that ${𝒞}_{r,s}(h+rs,c)=0\text{.}$ This implies $M(h+rs,c)$ is irreducible. Case (ii)b: Suppose ${ℒ}_{(h,c)}$ passes through one integer point $(r,s)$ with $rs<0\text{.}$ Since ${𝒞}_{r,s}(h,c)={𝒞}_{-r,s}(h+rs,c),$ the point $(-r,s)$ is on the line ${ℒ}_{(h,c)}$ (if we choose the line ${ℒ}_{(h,c)}$ carefully out of the four possible lines.) Also, $(-r,s)$ is the only integer point on ${ℒ}_{(h-rs,c)}\text{.}$ Then $(h-rs,c)$ falls into case (ii)a. We do not provide a proof of cases (hi) and (iv). The proof of these cases can be found in [FFu1990], Part II, Section 1. However, we do make a few comments to show that these results are reasonable. Case (iii): Suppose ${ℒ}_{(h,c)}$ passes through infinitely many integer points $(r_i,s_i)$ and crosses an axis at an integer point. We will assume the slope $\mu$ is positive and ${ℒ}_{(h,c)}$ passes through a the point $(0,s_0)$ for some $s_0\in {\mathbb{Z}}_{>0}\text{.}$ (If $\mu <0$ or ${ℒ}_{(h,c)}$ crosses the axis at a different point, we can still make arguments similar to those below.) Write $s_0=kp+\overline{s_0}$ where $k\in {\mathbb{Z}}_{\ge 0}$ and $0\ge \overline{s_0}<$ p\text{.}$$ We observe that $(r_1,s_1)=(-(k+1)q,\overline{s_0}-p)\text{.}$ (If $\overline{s_0}=0,$ then there are two points on the line ${ℒ}_{(h,c)},$ $(-(k+1)q,-p)$ and $(q,(k+1)p),$ with the same product. In this case, there is not a unique choice for $(r_1,s_1)\text{.}$ We choose either point.) We have $M(h+r_1{s}_1,c)\subseteq M(h,c),$ and $(r_1,-s_1)$ is on the line ${ℒ}_{(h+r_1{s}_1,c)}$ (for a careful choice of this line). The line ${\stackrel{\sim }{ℒ}}_{(h,c)}={ℒ}_{(h+r_1{s}_1,c)}$ so passes through infinitely many integer points and crosses an axis at an integer point, so that we can use the same arguments as above. We see that $({\stackrel{\sim }{r}}_1,{\stackrel{\sim }{s}}_1)=(-(k+2)q,-\overline{s_0}),$ which implies $M(h+r_1{s}_1+{\stackrel{\sim }{r}}_1{\stackrel{\sim }{s}}_1,c)\subseteq M(h+r_1{s}_1,c)\subseteq M(h,c)\text{.}$ Since $r_1{s}_1+{\stackrel{\sim }{r}}_1{\stackrel{\sim }{s}}_1=r_2{s}_2,$ we have $M(h+r_2{s}_2,c)\subseteq M(h+r_1{s}_1,c)\subseteq M(h,c)\text{.}$ We can continue this argument to get $M(h,c)\supseteq M(h+r_1{s}_1,c)\supseteq M(h+r_2{s}_2,c)\supseteq M(h+r_3{s}_3)\subseteq \cdots ,$ To show that $M(h,c)\subseteq M(h+r_{-1}{s}_{-1},c)\subseteq M(h+r_{-2}{s}_{-2},c)\subseteq \cdots ,$ we use the fact that ${𝒞}_{r,s}(h,c)={𝒞}_{-r,s}(h+rs,c)$ and apply the above argument to the Verma modules $M(h+r_{-i}{s}_{-i},c)\text{.}$ Therefore, the Verma module embeddings for $M(h,c)$ are at least those indicated in Theorem 3.5.1. Case (iv): We again have $M(h+r_1{s}_1,c)\subseteq M(h,c)\text{.}$ Since ${ℒ}_{(h,c)}$ does not cross at axis at an integer point, $\text{det}\hspace{0.17em}M{(h+r_1{s}_1,c)}^{(h+r_2{s}_2,c)}\ne 0\text{.}$ Therefore, $M{(h+r_1{s}_1,c)}^{(h+r_2{s}_2,c)}\cap M{(h,c)}_j=0$ for $j>0\text{.}$ However, $\text{ord}\hspace{0.17em}M{(h+t,c)}^{(h+t+r_2{s}_2,c)}=p(r_2{s}_2-r_1{s}_1)+1=\text{dim}\hspace{0.17em}M{(h+r_1{s}_1,c)}^{(h+r_2{s}_2,c)}+1\text{.}$ Using Theorem 2.5.1, we see that there must be some vector $0\ne v\in M{(h,c)}_1^{(h+r_2{s}_2,c)}$ so that $v\ne M(h+r_1{s}_1,c)\text{.}$ It remains to show that $v$ is a highest weight vector. $\square$

Another Description of Blocks

We can use the line ${ℒ}_{(h,c)}$ to generate lines corresponding to the entire block $\left[(h,c)\right]$ in the following way. If $(r,s)$ is an integer point on the line ${ℒ}_{(h,c)}$ let ${\stackrel{\sim }{ℒ}}_{(h,c)}$ be the line with the same slope as ${ℒ}_{(h,c)}$ and passing through the point $(-r,s)\text{.}$ Then ${\stackrel{\sim }{ℒ}}_{(h,c)}={\stackrel{\sim }{ℒ}}_{(h+rs,c)}$ corresponds to the weight $(h+rs,c)\in \left[(h,c)\right]\text{.}$ Using this approach we can construct a set of lines corresponding to the weights in a given block. In this section, we begin with a line and generate the weights in a given block.

Let ${ℒ}^{(\mu ,a,b)}$ be a line where $\mu$ is the slope of the line and $(a,b)$ is a point on the line. Then ${ℒ}^{(\mu ,a,b)}$ determines a weight $(h,c)$ by $$h=\frac{{(a\mu -b)}^2-{(\mu -1)}^2}{4\mu },c=13-6(\mu +\frac{1}{\mu })\text{.}$$ We will write $\left[(\mu ,a,b)\right]$ for $\left[(h,c)\right]$ if ${ℒ}^{(\mu ,a,b)}$ determines $(h,c)\text{.}$

•

Blocks of size two are indexed by triples $$\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℝ-\mathbb{Q}\hspace{0.17em}\text{with}\hspace{0.17em}\mid \mu \mid <1\hspace{0.17em}\text{and}\hspace{0.17em}a,b\in {\mathbb{Z}}_{>0}\}\cup \left\{(\mu ,a,a)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℂ-\mathbb{Q},\mid \mu \mid =1,a\in {\mathbb{Z}}_{>0}\}\text{.}$$ The weights in a block of size two $\left[(\mu ,a,b)\right]$ are indexed by triples $\left\{(\mu ,a,\pm b)\right\}\text{.}$

•

Infinite blocks with a maximal element are indexed by triples $$\left\{(\frac{p}{q},a,b)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{c}p,q\in {\mathbb{Z}}_{>0}\text{, with}\hspace{0.17em}\text{gcd}(p,q)=1,p<q\\ \text{If}\hspace{0.17em}2\nmid q\text{, then}\hspace{0.17em}0\le a<\frac{q}{2},\hspace{0.17em}0\le b<p\\ \text{If}\hspace{0.17em}2|q\text{, then}\hspace{0.17em}0\le a<q,\hspace{0.17em}0\le b<\frac{p}{2}\end{array}\}\text{.}$$ Infinite blocks with a minimal element are indexed by triples $$\left\{(-\frac{p}{q},-a,b)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{c}p,q\in {\mathbb{Z}}_{>0}\text{, with}\hspace{0.17em}\text{gcd}(p,q)=1,p<q\\ \text{If}\hspace{0.17em}2\nmid q\text{, then}\hspace{0.17em}0\le a<\frac{q}{2},\hspace{0.17em}0\le b<p\\ \text{If}\hspace{0.17em}2|q\text{, then}\hspace{0.17em}0\le a<q,\hspace{0.17em}0\le b<\frac{p}{2}\end{array}\}\text{.}$$ For a block $\left[(\pm \frac{p}{q},\pm a,b)\right]$ with $a\ne 0,$ the weights in the block are indexed by triples $$\left\{(\frac{p}{q},a,\pm b+2kp)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}\text{or}\left\{(-\frac{p}{q},-a,\pm b+2kp)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}\text{.}$$ For a block $\left[(\pm \frac{p}{q},0,b)\right],$ the weights in the block are indexed by triples $$\{(\frac{p}{q},0,b),(\frac{p}{q},0,\pm b+2kp)\hspace{0.17em}|\hspace{0.17em}k\in {\mathbb{Z}}_{>0}\}\text{or}\{(-\frac{p}{q},0,b),(-\frac{p}{q},0,\pm b+2kp)\hspace{0.17em}|\hspace{0.17em}k\in {\mathbb{Z}}_{>0}\}\text{.}$$

		Proof.
	Suppose $(\mu ,a,b)\to (h,c)$ and $\mid \left[(h,c)\right]\mid >1\text{.}$ Then, ${ℒ}^{(\mu ,a,b)}$ must pass through at least one integer point. Therefore, we can restrict to triples $(\mu ,a,b)$ with $a,b\in \mathbb{Z}\text{.}$ We now consider what values of $\mu$ will determine real values for $h$ and $c\text{.}$ Note that $c=13-6(\mu +\frac{1}{\mu })\in ℝ$ only if $\mu \in ℝ$ or $\mu \in ℂ$ with $\mid \mu \mid =1\text{.}$ Suppose $\mu \in ℂ-ℝ$ with $\mid \mu \mid =1\text{.}$ Then $\mu =A+Bi=$ with $B\ne 0\text{.}$ It is straightforward to check that $h=\frac{{(a\mu -b)}^2-{(\mu -1)}^2}{4\mu }\in ℝ$ only if $a^2=b^2\text{.}$ Recall that $(\mu ,a,b),$ $(\mu ,-a,-b),$ $(\frac{1}{\mu },a,b),$ and $(\frac{1}{\mu },-a,-b)$ all determine to the same weight $(h,c)\text{.}$ Therefore, we restrict our attention to triples $(\mu ,a,b)$ with $\mu \in ℝ$ so that $0<\mid \mu \mid <1$ and $a\in {\mathbb{Z}}_{>0},$ $b\in {\mathbb{Z}}_{\ne 0}\text{;}$ or with $\mu \in ℂ$ so that $\mid \mu \mid =1,$ $a\in {\mathbb{Z}}_{>0},$ and $b=\pm a\text{.}$ • We first consider blocks of size two. A pair $(h,c)$ belonging to a block of size two lies on exactly one curve ${𝒞}_{r,s}(h,c)=0,$ and so any line determining the pair $(h,c)$ passes through exactly one integer point. Therefore, triples in the set $\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℝ-\mathbb{Q}\hspace{0.17em}\text{with}\hspace{0.17em}\mid \mu \mid <1\hspace{0.17em}\text{and}\hspace{0.17em}a\in {\mathbb{Z}}_{>0},b\in \mathbb{Z}\}\cup \left\{(\mu ,a,\pm a)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℂ-\mathbb{Q},\mid \mu \mid =1,a\in {\mathbb{Z}}_{>0}\}$ are in one-to-one correspondence with such pairs $(h,c)\text{.}$ $$If $(h,c)$ is the pair defined by $(\mu ,a,b),$ then $M(h+ab,c)\subseteq M(h,c)\text{.}$ This implies that $(h+ab,c)$ corresponds to the line with slope $\mu$ passing through the point $(-a,b)\text{.}$ Therefore, any block of size two can be identified with a set $\left\{(\mu ,a,\pm b)\right\},$ with $\mu ,$ $a,$ and $b$ as in the previous paragraph. Taking one triple from each of these pairs of triples, we see that the set $$\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℝ-\mathbb{Q}\hspace{0.17em}\text{with}\hspace{0.17em}\mid \mu \mid <1\hspace{0.17em}\text{and}\hspace{0.17em}a,b\in {\mathbb{Z}}_{>0}\}\cup \left\{(\mu ,a,a)\hspace{0.17em}\right|\hspace{0.17em}\mu \in ℂ-\mathbb{Q},\mid \mu \mid =1,a\in {\mathbb{Z}}_{>0}\}$$ indexes the blocks of size two. • Now we consider infinite blocks. Let $(h,c)$ be a pair in an infinite block. We assume $0<\mu \le 1$ and $\mu \in \mathbb{Q}\text{.}$ (The arguments for $1\le \mu <0$ are the similar.) Write $\mu =\frac{p}{q}$ such that $p$ and $q$ are relatively prime. Consider weights $(h,c)$ which are maximal in their own block $\left[(h,c)\right]\text{.}$ Since $M(h,c)$ does not embed in any other Vermas, any line determined by $(h,c)$ must pass through only integer points $(a,b)$ such that $ab>0\text{.}$ It is clear that the triples corresponding to maximal weights are contained in the set $\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}0\le a<\frac{q}{2},0\le b<p\}$ (or $\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}0\le a<q,0\le b<\frac{p}{2}\}$ if $q$ is even). However, $(\frac{p}{q},a,b)$ and $(\frac{p}{q},q-a,p-b)$ determine the same weight. Therefore, the set $\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}0\le a<\frac{q}{2},0\le b<p\}$ (or $\left\{(\mu ,a,b)\hspace{0.17em}\right|\hspace{0.17em}0\le a<q,0\le b<\frac{p}{2}\}\text{)}$ contains exactly one triple corresponding to each such pair $(h,c)\text{.}$ $$We can also describe the block $\left[(h,c)\right]\text{.}$ Let $(\mu ,a,b)$ $\text{(}\mu =\frac{p}{q}\in {\mathbb{Q}}_{\ne 0}$ with $\mid \mu \mid \le 1$ and $a,b\in \mathbb{Z}$ with $0\le a<\frac{q}{2}$ be a triple which determines $(h,c)\text{.}$ Then the integer points lying on ${ℒ}^{(\frac{p}{q},a,b)}$ are $(a+kq,b+kp),$ $k\in \mathbb{Z}\text{.}$ This implies that $M(h+(a+kq)(b+kp),c)\subseteq M(h,c)\text{.}$ Therefore, the line given by $(\mu ,-a+kq,-(b+kp))$ must determine $(h+(a+kq)(b+kp),c)\in ℬ\text{.}$ This may not produce all pairs $(h,c)$ in the block (as in case (iv) of Theorem 3.5.1). Therefore, we also consider the pair $(h+ab,c)\in ℬ,$ which is determined by the triple $(\mu ,a,-b)\text{.}$ Using the same argument as above, we get the triples $(\mu ,a,-b)\text{.}$ Therefore, the set $$\left\{(\frac{p}{q},a-kq,\pm b+kp)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}⟷\left\{(\frac{p}{q},a,\pm b+2kp)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}$$ is in general a set of representatives for the elements of the block corresponding to $(\frac{p}{q},a,b)\text{.}$ If $a=0,$ the triples $(\frac{p}{q},0,b+2kp)$ and $(\frac{p}{q},0,-b-2kp)$ correspond to distinct lines but still determine the same weight. In this case, the set $\left\{(\frac{p}{q},0,\pm b+2kp)\hspace{0.17em}\right|\hspace{0.17em}k\in {\mathbb{Z}}_{\ge 0}\}$ forms set of representatives of the elements of the block. $$Consider the example with $\mu =\frac{2}{3}\text{.}$ $$\begin{array}{c} s=8 s=4 s=0 s=-4 ℒ(23,-2,1) ℒ(23,1,1) ℒ(23,1,-1) ℒ(23,4,-3) (-2,-1) (1,1) (4.3) \end{array}$$ The set of integer points $\{(a,b)\in {\mathbb{Z}}^2\hspace{0.17em}|\hspace{0.17em}0\le a<\frac{3}{2},0\le b<2\}$ indexes the infinite blocks with $c=13-6(\frac{2}{3}+\frac{3}{2})=0\text{.}$ The line ${ℒ}^{(\frac{2}{3},1,1)}$ determines the weight $(0,0)\text{.}$ From the integer points $(1,1),$ $(-2,-1),$ and $(4,3)$ on the line ${ℒ}^{(\frac{2}{3},1,1)},$ we get the lines ${ℒ}^{(1,-1)},$ ${ℒ}^{(\frac{2}{3},-2,1)},$ and ${ℒ}^{(\frac{2}{3},4,-3)}\text{;}$ these lines determine the weights $(1,0),$ $(2,0),$ and $(12,0)$ respectively. In general, the set of points $\left\{(1,4k\pm 1)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}$ correspond to the block $$\left\{(\frac{{(12k+2\pm 3)}^2-1}{24},0)\hspace{0.17em}\right|\hspace{0.17em}k\in \mathbb{Z}\}=\left\{(\frac{j(3j\pm 1)}{2},0)\hspace{0.17em}\right|\hspace{0.17em}j\in {\mathbb{Z}}_{\ge 0}\}\text{.}$$ $\square$

Define the group $W=⟨s_0,s_1\hspace{0.17em}|\hspace{0.17em}{s}_i^2=1⟩\text{.}$ We can define an action of $W$ on the triples $(\mu ,a,b)$ so that

(i)	a block of size two $\left[(\mu ,a,b)\right]$ is the orbit of the subgroup $⟨s_0⟩\subseteq W\text{;}$
(ii)	a infinite block $\left[(\pm \frac{p}{q},\pm a,b)\right],$ with $a\ne 0,$ is the orbit of $W\text{;}$
(iii)	a infinite block $\left[(\pm \frac{p}{q},0,b)\right],$ $b\ne 0,$ is the orbit of the subgroup $⟨s_1,s_0{s}_1{s}_0⟩\subseteq W\text{;}$
(iv)	a infinite block $\left[(\pm \frac{p}{q},0,0)\right]$ is of the form $\left\{{\left(s_1{s}_0\right)}^k(\pm \frac{p}{q},0,0)\hspace{0.17em}\right|\hspace{0.17em}k\in {\mathbb{Z}}_{\ge 0}\}\text{.}$

		Proof.
	We define an action of $W$ on triples $(\mu ,a,b)$ as follows: • $s_0$ is the reflection about 0: $s_0(\mu ,a,b)=(\mu ,a,-b)\text{;}$ • for $\mu =\frac{p}{q},$ $s_1$ is the reflection about $p\text{:}$ $s_1(\pm \frac{p}{q},\pm a,b)=(\pm \frac{p}{q},\pm a,-(b-p)+p)=(\pm \frac{p}{q},\pm a,-b+2p)\text{.}$ Then (i) and (ii) follow from the previous proposition. For (iii), note that $s_0{s}_1{s}_0$ is the reflection about $-p\text{.}$ Also, $(\frac{p}{q},0,-b+2kp)$ and $(\frac{p}{q},0,b-2kp)$ determine the same weight. Then $⟨s_1,s_0{s}_1{s}_0⟩$ generates $\left[(\pm \frac{p}{q},0,b)\right],$ where we replace $(\frac{p}{q},0,-b+2kp)$ with $(\frac{p}{q},0,b-2kp)$ for $k$ even. Finally, we have that $s_1{s}_0$ is translation by $2p\text{.}$ Then, (iv) follows. $\square$

Translation Functors

We now consider $M(h,c)\otimes L(h\prime ,c\prime )\text{.}$ From Theorem 2.3.12, we know that $$M(h,c)\otimes L(h\prime ,c\prime )=\underset{\left[\mu \right]\in \left[{𝔥}^{*}\right]}{⨁}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}$$ and ${(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\ne 0$ only if $\left[\mu \right]=\left[(h+h\prime +k,c+c\prime )\right]$ for some $k\in {\mathbb{Z}}_{\ge 0}\text{.}$ Moreover, we know this submodule has a filtration by Verma modules. In this section, we use the contravariant form to better describe ${(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\text{.}$

Recall $⟨,⟩:M(h,c)\otimes L(h\prime ,c\prime )\times M(h,c)\otimes L(h\prime ,c\prime )\to ℂ$ is defined by $$⟨v\otimes w,v\prime \otimes w\prime ⟩=⟨v,v\prime ⟩⟨w,w\prime ⟩$$ where $v,v\prime \in M(h,c)$ and $w,w\prime \in L(\stackrel{\sim }{h},c\prime )\text{.}$ This form on $M(h,c)\otimes L(h\prime ,c\prime )$ is contravariant.

Let $(h,c),(h\prime ,c\prime )\in {ℝ}^2,$ and let $\left\{w_{k,j}\hspace{0.17em}\right|\hspace{0.17em}1\le j\le \text{dim}\left(L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\right)\}$ be a basis for $L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\text{.}$ From Lemma 2.6.2, the following sets are bases for ${(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}\text{:}$ $$\begin{array}{cc}{\displaystyle \{d_{-\lambda }{v}^{+}\otimes w_{k,i}\hspace{0.17em}|\hspace{0.17em}\mid \lambda \mid =n,1\le i\le \text{dim}\left(L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\right)\}\text{;}}& \text{(3.16)}\end{array}$$ $$\begin{array}{cc}{\displaystyle \left\{d_{-\lambda }(v^{+}\otimes w_{k,i})\hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid =n,1\le i\le \text{dim}\left(L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\right)\}\text{.}}& \text{(3.17)}\end{array}$$ We defined $$\text{det}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +k,c+c\prime )}≔\text{det}\left(⟨d_{-\lambda }{v}^{+}\otimes w_{m,j},d_{-{\lambda }^{*}}{v}^{+}\otimes w_{m\prime ,j\prime }⟩\right)$$ where the entries in the matrix are indexed over partitions $\lambda$ and ${\lambda }^{*}$ and positive integers $m,m\prime ,j,j\prime$ such that $\mid \lambda \mid =k-m,$ $\mid {\lambda }^{*}\mid =k-m\prime ,$ $1\le j\le \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +m,c\prime )},$ and $1\le j\prime \le \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +m\prime ,c\prime )}\text{.}$ From Lemma 2.6.3, we have $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \text{det}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +k,c+c\prime )}}& =& {\displaystyle \prod _{j\le k}{\left(\text{det}\hspace{0.17em}M{(h,c)}^{(h+k-j,c)}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}}\\ & & {\displaystyle \times {\left(\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{h\prime +j,c\prime }\right)}^{p(k-j)}\text{.}}\end{array}}& \text{(3.18)}\end{array}$$

For $(h,c),(h\prime ,c\prime )\in {ℝ}^2$ and $k\in {\mathbb{Z}}_{\ge 0},$ $$\text{det}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +k,c+c\prime )}$$ is given by $$\prod _{0\le j\le k}{\left(\text{det}\hspace{0.17em}M{(h+h\prime +j,c+c\prime )}^{(h+h\prime +k,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}{\left(a_j^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}\right)}^{p(k-j)},$$ where $$a_j^{(h\prime ,c\prime )}(h,c)=\prod _{\underset{rs\le j}{1\le r\le s}}{\left(\frac{{𝒞}_{r,s}(h,c)}{{𝒞}_{r,s}(h+h\prime +j-rs,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j-rs,c\prime )}}\text{.}$$

		Proof.
	Using Equation 3.18, we only need to show that $$\prod _{j\le k}{\left(a_j^{(h\prime ,c\prime )}(h,c)\right)}^{p(k-j)}=\prod _{j\le k}{\left(\frac{\text{det}\hspace{0.17em}M{(h,c)}^{(h+k-j,c)}}{\text{det}\hspace{0.17em}M{(h+h\prime +j,c+c\prime )}^{(h+h\prime +k,c+c\prime )}}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}\text{.}$$ Note that $$\begin{array}{lll}\multicolumn{3}{c}{{\displaystyle \prod _{0\le j\le k}{\left(\frac{\text{det}\hspace{0.17em}M{(h,c)}^{(h+k-j,c)}}{\text{det}\hspace{0.17em}M{(h+h\prime +j,c+c\prime )}^{(h+h\prime +k,c+c\prime )}}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}}}\\ & =& {\displaystyle \prod _{\underset{1\le r\le s}{0\le j\le k}}{\left(\frac{{𝒞}_{r,s}(h,c)}{{𝒞}_{r,s}(h+h\prime +j,c+c\prime )}\right)}^{p(k-j-rs)\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}}\\ & =& {\displaystyle \underset{\underset{\underset{p(k-j-rs)=0\hspace{0.17em}\text{for}\hspace{0.17em}j>k\hspace{0.17em}\text{and}\hspace{0.17em}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}=0\hspace{0.17em}\text{for}\hspace{0.17em}j<0\text{.}}{\text{We can let the product range over all}\hspace{0.17em}j\in \mathbb{Z}\hspace{0.17em}\text{since}}}{⏟}}{\prod _{\underset{1\le r\le s}{j\in \mathbb{Z}}}{\left(\frac{{𝒞}_{r,s}(h,c)}{{𝒞}_{r,s}(h+h\prime +j,c+c\prime )}\right)}^{p(k-j-rs)\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}}}\\ & =& {\displaystyle \underset{\underset{\text{We shift}\hspace{0.17em}j\to j-rs}{⏟}}{\prod _{\underset{1\le r\le s}{j\in \mathbb{Z}}}{\left(\frac{{𝒞}_{r,s}(h,c)}{{𝒞}_{r,s}(h+h\prime +j-rs,c+c\prime )}\right)}^{p(k-j)\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j-rs,c\prime )}}}}\\ & =& {\displaystyle \prod _{\underset{\underset{rs\le j}{1\le r\le s}}{0\le j\le k}}{\left(\frac{{𝒞}_{r,s}(h,c)}{{𝒞}_{r,s}(h+h\prime +j-rs,c+c\prime )}\right)}^{p(k-j)\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j-rs,c\prime )}}\text{.}}\end{array}$$ $\square$

Fix $(h\prime ,c\prime )\in {ℝ}^2\text{.}$ Consider $(h,c)\in {ℝ}^2$ such that $(h+n,c)\notin \left[(h,c)\right]$ for any $n\in {\mathbb{Z}}_{>0}$ (i.e. $M(h,c)$ is irreducible). For each $\left[\mu \right]\in \left[{𝔥}^{*}\right]$ and $n\in {\mathbb{Z}}_{\ge 0},$ there is a projection map $${\text{Pr}}_n^{\left[\mu \right]}:{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}\to {\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$$ given by $${\text{Pr}}_n^{\left[\mu \right]}\left(w\right)=w-\sum _{\left[\gamma \right]\ne \left[\mu \right]}\sum _1^m⟨w,v_i^{\left[\gamma \right]}⟩{\bar{v}}_i^{\left[\gamma \right]}$$ where $\{v_1^{\left[\gamma \right]},\dots ,v_m^{\left[\gamma \right]}\}$ and $\{{\bar{v}}_1^{\left[\gamma \right]},\dots ,{\bar{v}}_m^{\left[\gamma \right]}\}$ are dual bases for ${\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}\text{.}$

		Proof.
	Since $(h+n,c)\notin \left[(h,c)\right]$ for all $n\in {\mathbb{Z}}_{\ge 0},$ Equation 3.18 implies that the contravariant form is nondegenerate on ${(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}\text{.}$ From Proposition 2.7, distinct blocks are orthogonal with respect to the form. Therefore, the contravariant form is nondegenerate on each block. Let $\{v_1^{\left[\gamma \right]},\dots ,v_m^{\left[\gamma \right]}\}$ be a basis for ${\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}\text{.}$ Since the contravariant form is nondegenerate on this space, there is a dual basis $\{{\bar{v}}_1^{\left[\gamma \right]},\dots ,{\bar{v}}_m^{\left[\gamma \right]}\}$ for this space, i.e. $⟨v_i^{\left[\gamma \right]},{\bar{v}}_l^{\left[\gamma \right]}⟩={\delta }_{i,l}\text{.}$ We define a map $${\text{Pr}}_n^{\left[\mu \right]}:{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}\to {(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}$$ by $${\text{Pr}}_n^{\left[\mu \right]}\left(w\right)=w-\sum _{\left[\gamma \right]\ne \left[\mu \right]}\sum _1^m⟨w,v_i^{\left[\gamma \right]}⟩{\bar{v}}_i^{\left[\gamma \right]}\text{.}$$ Note that $⟨{\text{Pr}}_n^{\left[\mu \right]}\left(w\right),v_i^{\left[\gamma \right]}⟩=0$ whenever $\left[\gamma \right]\ne \left[\mu \right]$ since distinct blocks are orthogonal. Therefore, ${\text{Pr}}_n^{\left[\mu \right]}\left(w\right)\in {\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}\text{.}$ Also, for $w\in {\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}\text{,}$ ${\text{Pr}}_n^{\left[\mu \right]}\left(w\right)=w\text{.}$ $\square$

Fix $(h\prime ,c\prime )\in {ℝ}^2$ and $n\in {\mathbb{Z}}_{\ge 0}\text{.}$ Suppose $(h,c)\in {ℝ}^2$ is such that $(h+j,c)\notin \left[(h,c)\right]$ and $(h+h\prime +j,c+c\prime )\notin \left[(h+h\prime +k,c+c\prime )\right]$ for all $j,k\le n$ with $j\ne k\text{.}$
$$Then the submodule of $M(h,c)\otimes L(h\prime ,c\prime )$ generated by $$\underset{0\le j\le n}{⨁}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +j,c+c\prime )}$$ is isomorphic to $$\underset{0\le j\le n}{⨁}M{(h+h\prime +j,c+c\prime )}^{\oplus \text{dim}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}\text{.}$$ For a suitable choice of generating highest weight vectors $\left\{v_{j,i}^{+}\hspace{0.17em}\right|\hspace{0.17em}1\le i\le \text{dim}{\left(L(h\prime ,c\prime )\right)}^{(h\prime +j,c\prime )}\}$ of $\underset{0\le j\le n}{⨁}M{(h+h\prime +j,c+c\prime )}^{\oplus \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}\subseteq M(h,c)\otimes L(h\prime ,c\prime ),$ this sum is orthogonal with respect to the contravariant form on $M(h,c)\otimes L(h\prime ,c\prime ),$ and $$\prod _{1\le i\le \text{dim}{\left(L(h\prime ,c\prime )\right)}^{(h\prime +j,c\prime )}}⟨v_{j,i}^{+},v_{j,i}^{+}⟩=a_j^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}\text{.}$$

		Proof.
	Since $\left[(h,c)\right]\ne \left[(h+j,c)\right]$ for all $j\le n,$ the projection maps from the previous lemma are well-defined. We have assumed $\left[(h+h\prime +j,c+c\prime )\right]\ne \left[(h+h\prime +k,c+c\prime )\right]$ for $j,k\le n$ and $j\ne k\text{.}$ Therefore, for ${\mu }_j=(h+h\prime +j,c+c\prime )$ with $j\le n,$ the set $\left\{{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,i})\hspace{0.17em}\right|\hspace{0.17em}1\le i\le \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}\}$ is a basis for ${\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[{\mu }_j\right]}\right)}^{(h+h\prime +j,c+c\prime )},$ made up of highest weight vectors. Choose vectors $\left\{v_{j,i}^{+}\right\}$ such that • the transition matrix from $\left\{{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,i})\right\}$ to $\left\{v_{j,i}^{+}\right\}$ has determinant 1; • $⟨v_{j,i}^{+},v_{j,k}^{+}⟩=0$ if $i\ne k\text{.}$ Note that $$\prod _i⟨v_{j,i}^{+},v_{j,i}^{+}⟩=\text{det}\left(⟨{\text{Pr}}_j^{\left[{\mu }_{j}j\right]}(v^{+}\otimes w_{j,i}),{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,k})⟩\right)\text{.}$$ Then $\text{det}\left(⟨d_{-\lambda }{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,i}),d_{-\stackrel{\sim }{\mu }}{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,k})⟩\right)=\prod _i\left(⟨d_{-\mu }{v}_{j,i}^{+},d_{-\stackrel{\sim }{\lambda }}{v}_{j,i}^{+}⟩\right)$ is $${\left(\prod _i⟨v_{j,i}^{+},v_{j,i}^{+}⟩\right)}^{p(n-j)}\times {\left(\text{det}\hspace{0.17em}{(h+h\prime +j,c+c\prime )}^{(h+h\prime +n,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}\text{.}$$ Therefore, we only need to determine $\prod _i⟨v_{j,i}^{+},v_{j,i}^{+}⟩\text{.}$ We do this inductively. Suppose $$\text{det}\left(⟨{\text{Pr}}_k^{\left[{\mu }_k\right]}(v^{+}\otimes w_{k,i}),{\text{Pr}}_k^{\left[{\mu }_k\right]}(v^{+}\otimes w_{k,l})⟩\right)=a_k^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}$$ for $k<j\text{.}$ Since distinct blocks are orthogonal, we have $\text{det}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +j,c+c\prime )}$ is given by $$\begin{array}{cl}& {\displaystyle \prod _{k\le j}\text{det}\left(⟨d_{-\mu }{\text{Pr}}_j^{\left[{\gamma }_k\right]}(v^{+}\otimes d_{-\lambda }{w}^{+}),d_{-\stackrel{\sim }{\mu }}{\text{Pr}}_j^{\left[{\gamma }_k\right]}(v^{+}\otimes d_{-\stackrel{\sim }{\lambda }}{w}^{+})⟩\right)}\\ =& {\displaystyle \text{det}\left(⟨d_{-\mu }{\text{Pr}}_j^{\left[\mu \right]}(v^{+}\otimes d_{-\lambda }{w}^{+}),d_{-\stackrel{\sim }{\mu }}{\text{Pr}}_j^{\left[\mu \right]}(v^{+}\otimes d_{-\stackrel{\sim }{\lambda }}{w}^{+})⟩\right)}\\ & {\displaystyle \prod _{k<j}({\left(\text{det}\hspace{0.17em}M{(h+h\prime +k,c+c\prime )}^{(h+h\prime +j,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}}}\\ & {\displaystyle \times {\left(a_k^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\right)}^{p(k-j)})\text{.}}\end{array}$$ From Lemma 3.6.1, this implies $$\text{det}\left(⟨{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,i}),{\text{Pr}}_j^{\left[{\mu }_j\right]}(v^{+}\otimes w_{j,l})⟩\right)=a_j^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}\text{.}$$ $\square$

For $\gamma =(h+h\prime +k,c+c\prime ),$ the set $$B_n^{\left[\gamma \right]}=\left\{d_{-\mu }\left({\text{Pr}}_j^{\left[\gamma \right]}(v^{+}\otimes d_{-\lambda }{w}^{+})\right)\hspace{0.17em}\right|\hspace{0.17em}(h+h\prime +j,c+c\prime )\in \left[\gamma \right],\mid \lambda \mid =j,\mid \mu \mid =n-j\}$$ is a basis for ${\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}\text{.}$ Define $$\text{det}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}=\text{det}{\left(⟨v,w⟩\right)}_{v,w\in B_n^{\left[\gamma \right]}}$$

Let $(h,c),(h\prime ,c\prime )\in {ℝ}^2,$ $\left[\gamma \right]\in \left[{𝔥}^{*}\right],$ and $n\in {\mathbb{Z}}_{\ge 0}\text{.}$ Suppose $(h+k,c)\notin \left[(h,c)\right]$ for all $0\le k\le n\text{.}$ Then, $$\text{det}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}$$ is $$\begin{array}{cl}=& {\displaystyle \prod _{(h+h\prime +j,c+c\prime )\in \left[\gamma \right]}({\left(\text{det}\hspace{0.17em}M{(h+h\prime +j,c+c\prime )}^{(h+h\prime +n,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}}\\ & {\displaystyle \times {\left(a_j^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c)}\right)}^{p(n-j)})}\end{array}$$

		Proof.
	Let $n\in {\mathbb{Z}}_{\ge 0}$ and let $K$ be any set of positive integers between $0$ and $n\text{.}$ Fix $(h\prime ,c\prime )\in {ℝ}^2$ and consider all $(h,c)\in {ℝ}^2$ such that $\left[(h+h\prime +k,c+c\prime )\right]\ne \left[(h+h\prime +k\prime ,c+c\prime )\right]$ for any $k\prime$ such that $k\prime \notin K\text{.}$ Let $$M^K=\sum _{k\in K}{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +k,c+c\prime )}$$ We can construct projection maps ${\text{Pr}}_j^K:{(M(h,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +j,c+c\prime )}\to {\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}$ analogous to those in Propostion 3.6.2. Applying these projection maps to the basis to $\{v^{+}\otimes w_i^j\},$ we can construct a basis $\{v_1,\dots ,v_m\}$ for ${\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}$ which are linear combinations of the basis $\left\{d_{-\lambda }(v^{+}\otimes w_i^j)\hspace{0.17em}\right|\hspace{0.17em}j\le n,\mid \lambda \mid =n-j\}$ with coefficients which are rational functions of $h$ and $c\text{.}$ Consider $$\text{det}{\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}=\text{det}{\left(⟨v_i,v_j⟩\right)}_{1\le i,j\le m}\text{.}$$ This will be a rational function in $h$ and $c\text{.}$ For most choices of $(h,c),$ $\left[(h+h\prime +k,c+c\prime )\right]=\left\{(h+h\prime +k,c+c\prime )\right\}$ for each $k\in K$ and so $$M^K\cong \underset{k\in K}{⨁}M{(h+h\prime +k,c+c\prime )}^{\oplus \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}}\text{.}$$ Write ${\gamma }_k=(h+h\prime +k,c+c\prime )\text{.}$ Lemma 3.6.3 implies that for such choices of $h$ and $c,$ $$\begin{array}{rcll}{\displaystyle \text{det}{\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}}& =& {\displaystyle \prod _{k\in K}\text{det}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[{\gamma }_k\right]}\right)}^{(h+h\prime +n,c+c\prime )}}& \text{(3.19)}\\ {\displaystyle }& =& {\displaystyle \prod _{k\in K}{\left(a_k^{(h\prime ,c\prime )}(h,c)L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}\right)}^{p(n-k)}M{(h+h\prime +k,c+c\prime )}^{(h+h\prime +n,c+c\prime )}\text{.}}& \text{(3.20)}\end{array}$$ Since $\text{det}{\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}$ is a rational function of $h$ and $c,$ Equation 3.20 holds for all $(h,c)$ where $\text{det}{\left(M^K\right)}^{(h+h\prime +n,c+c\prime )}$ is defined. In particular, if $\left[\gamma \right]\cap \left\{(h+h\prime +j,c+c\prime )\hspace{0.17em}\right|\hspace{0.17em}0\le j\le n\}=\left\{(h+h\prime +k,c+c\prime )\hspace{0.17em}\right|\hspace{0.17em}k\in K\},$ then $\text{det}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}^{\left[\gamma \right]}\right)}^{(h+h\prime +n,c+c\prime )}$ is $$\prod _{j\in \left[\gamma \right]}{\left(a_j^{(h\prime ,c\prime )}(h,c)\hspace{0.17em}\text{det}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}\right)}^{p(n-j)}{\left(\text{det}\hspace{0.17em}M{(h+h\prime +j,c+c\prime )}^{(h+h\prime +n,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +j,c\prime )}}\text{.}$$ $\square$

We define a Jantzen-type filtration on $M\left(\lambda \right)\otimes L\left(\mu \right)$ in the following way. For an indeterminant $t,$ we define the Vir-module $M(h+t,c)$ as in Section 2.5. The map $\epsilon :ℂ\left[t\right]\to ℂ$ $(t\mapsto 0)$ to a map $$\epsilon :M(h+t,c)\otimes L(h\prime ,c\prime )⟶M(h,c)\otimes L(h\prime ,c\prime )\text{.}$$ For each $j\in {\mathbb{Z}}_{\ge 0},$ define $${(M(h+t,c)\otimes L(h\prime ,c\prime ))}_j=\{v\in M(h+t,c)\otimes L(h\prime ,c\prime )\hspace{0.17em}|\hspace{0.17em}{t}^j|⟨v,w⟩\hspace{0.17em}\text{for all}\hspace{0.17em}w\in M(h+t,c)\otimes L(h\prime ,c\prime )\}$$ and $${(M(h,c)\otimes L(h\prime ,c\prime ))}_j=\epsilon \left({(M(h+t,c)\otimes L(h\prime ,c\prime ))}_j\right)\text{.}$$

Let $j\in {\mathbb{Z}}_{\ge 0}\text{.}$ Then $${(M(h,c)\otimes L(h\prime ,c\prime ))}_j=M{(h,c)}_j\otimes L(h\prime ,c\prime )\text{.}$$

		Proof.
	Let $v\in {(M(h+t,c)\otimes L(h\prime ,c\prime ))}_j\text{.}$ Since distinct weight spaces are orthogonal with respect to the contravariant form, we may assume $v\in {(M(h+t,c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +t+n,c+c\prime )}$ for some $n\in {\mathbb{Z}}_{\ge 0}\text{.}$ For each $j\le n,$ let $\left\{w_{j,i}\right\}$ be a basis for $L(h\prime ,c\prime )$ which is orthonormal with respect to the contravariant form. (Such a basis exists since the contravariant form is nondegenerate on $L(h\prime ,c\prime )\text{.)}$ We may write $$v=\sum _{j=0}^n\sum _i{v}_{j,i}\otimes w_{j,i}$$ for some $v_{j,i}\in M(h+t,c)\text{.}$ Then, for any $v\prime \in M(h+t,c),$ $k\le n,$ and $1\le m\le \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )},$ $$\begin{array}{rcl}{\displaystyle ⟨v,v\prime \otimes w_{k,m}⟩}& =& {\displaystyle \sum _{j=0}^n\sum _i⟨v_{j,i}\otimes w_{j,i},v\prime \otimes w_{k,m}⟩}\\ & =& {\displaystyle \sum _{j=0}^n\sum _i⟨v_{j,i},v\prime ⟩⟨w_{j,i},w_{k,m}⟩}\\ & =& {\displaystyle ⟨v_{k,m},v\prime ⟩\text{.}}\end{array}$$ This implies $t^j|⟨v_{k,m},w⟩$ for all $w\in M(h,c)\otimes L(h\prime ,c\prime )$ and so $v_{k,m}\in M(h+t,c)\text{.}$ $\square$

Let $(h,c),(h\prime ,c\prime )\in {ℂ}^2$ and $\left[\mu \right]\in \left[{𝔥}^{*}\right]\text{.}$ Then, for each $n\in {\mathbb{Z}}_{\ge 0}$ $$\sum _{j>0}\text{dim}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}_j^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$$ is $$\text{ord}\left(\prod _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}{\left(a_k^{(h\prime ,c\prime )}(h+t,c)\right)}^{p(n-k)}\right)\text{.}$$

		Proof.
	From the previous lemma and Theorem 3.5.2, we know that for each $n\in {\mathbb{Z}}_{\ge 0}$ $$\sum _{j>0}\text{dim}{\left({(M(h,c)\otimes L(h\prime ,c\prime ))}_j\right)}^{(h+h\prime +n,c+c\prime )}$$ is given by $$\begin{array}{cc}{\displaystyle \text{ord}\hspace{0.17em}\prod _{0\le k\le n}{\left(\text{det}\hspace{0.17em}M{(h,c)}^{(h+t+n-k,c)}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k,c\prime )}}=\text{ord}\hspace{0.17em}\prod _{0\le k\le n}{\left(a_k^{(h\prime ,c\prime )}(h+t,c)\right)}^{p(n-k)}\text{.}}& \text{(3.21)}\end{array}$$ Therefore, to prove the result, we only need to show how these zeros are distributed. From the previous lemma, we have ${(M(h,c)\otimes L(h\prime ,c\prime ))}_j=M{(h,c)}_j\otimes L(h\prime ,c\prime )\text{.}$ Theorem 3.5.2 gives the structure of the Jantzen filtration for $M(h,c)\text{.}$ We will consider the cases of this result separately. Case (i): In this case, $M(h,c)$ is irreducible and so $M{(h,c)}_j=0$ for all $j\text{.}$ This corresponds to $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)}\ne 0$ for all $n\in {\mathbb{Z}}_{\ge 0},$ implying $\text{ord}\left(a_k^{(h\prime ,c\prime )}(h+t,c)\right)=0\text{.}$ Cases (ii) and (iii): There are integer points $(r_i,s_i),$ $1\le i\le k$ for some $k\in {\mathbb{Z}}_{>0},$ on the line ${ℒ}_{(h,c)}$ such that • $M{(h,c)}_j=M(h+r_j{s}_j,c)$ for $j\le k\text{;}$ • ${\left(M{(h,c)}_j\right)}^{(h+m,c)}=0$ for $j>k$ and $m\le n\text{.}$ Then we have a correspondence between • distinct zeros in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+m,c)},$ which will have the form ${𝒞}_{r_j,s_j}(h,c)\text{;}$ • $j$ such that ${\left(M{(h,c)}_j\right)}^{(h+m,c)}\ne 0\text{.}$ Moreover, the multiplicity of the zero ${𝒞}_{r_j,s_j}(h,c)$ in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+m,c)}$ is $p(m-r_j{s}_j)=\text{dim}\hspace{0.17em}M{(h+r_j{s}_j,c)}^{(h+m,c)}\text{.}$ Now, if $1\le j\le k,$ we can describe the decomposition of $M{(h,c)}_j\otimes L(h\prime ,c\prime )=M(h+r_j{s}_j,c)\otimes L(h\prime ,c\prime )$ by blocks. In particular, by Proposition 2.6.1, we know that ${(M(h+r_j{s}_j,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}$ has a filtration by Verma modules $0=M_0\subseteq M_1\subseteq \cdots$ such that • ${(M(h+r_j{s}_j,c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}=\bigcup M_i\text{;}$ • $M_i/M_{i-1}\cong M{(h+r_j{s}_j+h\prime +k_{j,i},c+c\prime )}^{\oplus \text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j,i},c\prime )}}$ for each $k_{j,i}\in {\mathbb{Z}}_{\ge 0}$ such that $(h+r_j{s}_j+h\prime +k_{j,i},c+c\prime )\in \left[\mu \right]\text{.}$ This means that $$\text{dim}{\left({(M{(h,c)}_j\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}=\sum _{k_{j,i}}p(n-(r_j{s}_j+k_{j,i}))\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j,i},c\prime )},$$ where we sum over $\left\{k_{j,i}\hspace{0.17em}\right|\hspace{0.17em}(h+r_j{s}_j+h\prime +k_{j,i},c+c\prime )\in \left[\mu \right]\}\text{.}$ Then, $\sum _j\text{dim}{\left({(M{(h,c)}_j\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$ is given by $$\begin{array}{cc}{\displaystyle \sum _j\sum _{k_{j,i}|(h+r_j{s}_j+h\prime +k_{j,i},c+c\prime )\in \left[\mu \right]}p(n-(r_j{s}_j+k_{j,i}))\hspace{0.17em}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j,i},c\prime )}\text{.}}& \text{(3.22)}\end{array}$$ On the other hand, $$\prod _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}{\left(a_k^{(h\prime ,c\prime )}(h+t,c)\right)}^{p(n-k)}$$ is $$\begin{array}{cc}{\displaystyle \text{ord}\left(\prod _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}\prod _{\underset{rs\le k}{1\le r\le s}}{\left(\frac{{𝒞}_{r,s}(h+t,c)}{{𝒞}_{r,s}(h+t+h\prime +k-rs,c+c\prime )}\right)}^{\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k-rs,c\prime )}p(n-k)}\right)\text{.}}& \text{(3.23)}\end{array}$$ Given the correspondence stated earlier, we see that (3.23) is $$\sum _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}\sum _{j|r_j{s}_j<k}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k-r_j{s}_j,c\prime )}p(n-k),$$ which is equal to (3.22). Case (iv): We have $$\begin{array}{cc}{\displaystyle M{(h,c)}_j=M(h+n_{j,1},c)+M(h+n_{j,2},c)}& \text{(3.24)}\end{array}$$ where $$\begin{array}{cc}{\displaystyle M(h+n_{j,1},c)\cap M(h+n_{j,2},c)=M{(h,c)}_{j+1}\text{.}}& \text{(3.25)}\end{array}$$ Consider $n_{j_0,i}$ maximal so that $n_{j_0,i}\le n\text{.}$ Then, $$\begin{array}{rcl}{\displaystyle {(M{(h,c)}_{j_0}\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}}& =& {\displaystyle {(M(h+n_{j_0,1},c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}}\\ & =& {\displaystyle \oplus {(M(h+n_{j_0,2},c)\otimes L(h\prime ,c\prime ))}^{(h+h\prime +n,c+c\prime )}\text{.}}\end{array}$$ Again, we know the decomposition of each of these summands by blocks. The module ${(M(h+n_{j_0,i},c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}$ has a filtration by Verma modules where $M(h+n_{j_0,i}+k_{j_0,i,l},c+c\prime )$ appears with multiplicity $\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j_0,i,l},c\prime )}$ for each $k_{j_0,i,l}$ such that $(h+n_{j_0,i}+k_{j_0,i,l},c+c\prime )\in \left[\mu \right]\text{.}$ Therefore, $\text{dim}{\left({(M{(h,c)}_{j_0}\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$ is $$\begin{array}{cl}=& {\displaystyle \text{dim}{\left({(M(h+n_{j_0,1},c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}}\\ & {\displaystyle +\text{dim}{\left({(M(h+n_{j_0,2},c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}}\\ =& {\displaystyle \sum _{k_{j_0,i,l}}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j_0,i,l},c\prime )}p(n-(n_{j_0,i}+k_{j_0,i,l}))\text{.}}\end{array}$$ Using (3.24) and (3.25), we can similarly argue that $\text{dim}{\left({(M{(h,c)}_{j_0-1}\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$ is $$\begin{array}{cl}=& {\displaystyle \text{dim}{\left({(M(h+n_{j_0-1,1},c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}}\\ & {\displaystyle +\text{dim}{\left({(M(h+n_{j_0-1,2},c)\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}}\\ & {\displaystyle -\text{dim}{\left({(M{(h+,c)}_{j_0}\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}}\\ =& {\displaystyle \sum _{k_{j_0,i,l}}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j_0-1,i,l},c\prime )}p(n-(n_{j_0-1,i}+k_{j_0-1,i,l}))}\\ & {\displaystyle -\sum _{k_{j_0-1,i,l}}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j_0,i,l},c\prime )}p(n-(n_{j_0,i}+k_{j_0,i,l}))\text{.}}\end{array}$$ In general, $$\text{dim}{\left({(M{(h,c)}_{j_0-m}\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$$ is given by $$\sum _{s=0}^m{(-1)}^{m-s}\sum _{k_{j_0-s,i,l}}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{j_0-s,i,l},c\prime )}p(n-(n_{j_0-s,i}+k_{j_0+s,i,l}))\text{.}$$ Suppose that $n_{j,i}\le n$ for $j\le m$ and $n_{j,i}>n$ for $j>m\text{.}$ (It may be the case that $n_{j,i}\le n$ and $n_{j,2}>n\text{.}$ However, the same argument works with only minor modifications.) Then, $$\sum _{j\in {\mathbb{Z}}_{>0}}\text{dim}{\left({(M{(h,c)}_j\otimes L(h\prime ,c\prime ))}^{\left[\mu \right]}\right)}^{(h+h\prime +n,c+c\prime )}$$ is $$\begin{array}{cc}{\displaystyle \sum _{s=0}^{\lfloor \frac{m-1}{2}\rfloor }\sum _{k_{2s+1,i,l}}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k_{2s+1,i,l},c\prime )}p(n-(n_{2s+1,i}+k_{2s+1,i,l}))}& \text{(3.26)}\end{array}$$ Again, the distinct zeros in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+m,c)}$ will be exactly of the form ${𝒞}_{r_{j,i},s_{j,i}}(h,c),$ where $j=2s+1,$ $0\le s\le \lfloor \frac{m-1}{2}\rfloor ,$ and $r_{j,i}{s}_{j,i}=n_{j,i}\text{.}$ Moreover, the multiplicity of the zero ${𝒞}_{r_{j,i},s_{j,i}}(h,c)$ in $\text{det}\hspace{0.17em}M{(h,c)}^{(h+m,c)}$ is $p(m-n_{j,i})=\text{dim}\hspace{0.17em}M{(h+n_{j,i},c)}^{(h+m,c)}\text{.}$ We then see that $$\text{ord}\left(\prod _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}{\left(a_k^{(h\prime ,c\prime )}(h+t,c)\right)}^{p(n-k)}\right)$$ is $$\sum _{\underset{(h+h\prime +k,c+c\prime )\in \left[\mu \right]}{0\le k\le n}}\sum _{j,i|n_{j,i}<k}\text{dim}\hspace{0.17em}L{(h\prime ,c\prime )}^{(h\prime +k-n_{j,i},c\prime )}p(n-k),$$ which is equal to (3.26). $\square$

Notes and References

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

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