Last updated: 26 March 2015

This is an excerpt of the paper *Fusion Product of Positive Level Representations and Lie Algebra Homology* by Shrawan Kumar,
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA.

Let $\U0001d524$ be a finite dimensional complex simple Lie algebra. (We also fix a Borel subalgebra $\U0001d51f$ and a Cartan subalgebra
$\U0001d525\subset \U0001d51f$ of $\U0001d524\text{.)}$ Then the associated *affine Kac-Moody Lie algebra*
is by definition the space
$$\stackrel{\sim}{\U0001d524}\u2254\U0001d524{\otimes}_{\u2102}\u2102\left[{t}^{\pm 1}\right]\oplus \u2102K$$
together with the Lie bracket (for $X,Y\in \U0001d524$ and $P,Q\in \u2102\left[{t}^{\pm 1}\right]\text{)}$
$$\begin{array}{c}[X\otimes P,Y\otimes Q]=[X,Y]\otimes PQ+\left(\u27e8X,Y)\underset{t=0}{\text{Res}}\left(\frac{dP}{dt}Q\right)\right)K,\phantom{\rule{1em}{0ex}}\text{and}\\ [\stackrel{\sim}{\U0001d524},K]=0,\end{array}$$
where $\u27e8\xb7,\xb7)$ is the Killing form on $\U0001d524,$
normalized so that $\u27e8\theta ,\theta \u27e9=2$
for the highest root $\theta $ of $\U0001d524\text{.}$

The Lie algebra $\U0001d524$ sits as a Lie subalgebra of $\stackrel{\sim}{\U0001d524}$ as $\U0001d524\otimes {t}^{0}\text{.}$ The Lie algebra $\stackrel{\sim}{\U0001d524}$ admits a distinguished "parabolic" subalgebra $$\stackrel{\sim}{\U0001d52d}\u2254\U0001d524\otimes \u2102\left[t\right]\oplus \u2102K\text{.}$$ We also define its "nil-radical" $\stackrel{\sim}{\U0001d532}$ (which is an ideal of $\stackrel{\sim}{\U0001d52d}\text{)}$ by $$\stackrel{\sim}{\U0001d532}\u2254\U0001d524\otimes t\u2102\left[t\right],$$ and its "Levi component" (which is a Lie subalgebra of $\stackrel{\sim}{\U0001d52d}\text{)}$ $${\stackrel{\sim}{\U0001d52d}}^{0}\u2254\U0001d524\otimes {t}^{0}\oplus \u2102K\text{.}$$ Clearly (as a vector space) $$\stackrel{\sim}{\U0001d52d}=\stackrel{\sim}{\U0001d532}\oplus {\stackrel{\sim}{\U0001d52d}}^{0}\text{.}$$

Also define ${\stackrel{\sim}{\U0001d532}}^{-}=\U0001d524\otimes {t}^{-1}\u2102\left[{t}^{-1}\right]\subseteq \stackrel{\sim}{\U0001d524},$ and the Cartan subalgebra $\stackrel{\sim}{\U0001d525}=\U0001d525\otimes {t}^{0}\oplus \u2102K$ of $\stackrel{\sim}{\U0001d524}\text{.}$ Let $W$ (resp. $\stackrel{\sim}{W}\text{)}$ be the Weyl group of $\U0001d524$ (resp. $\stackrel{\sim}{\U0001d524}\text{).}$ Then $\stackrel{\sim}{W}$ acts on the dual space ${\stackrel{\sim}{\U0001d524}}^{*}$ by linear automorphisms. Let $\rho \in {\U0001d525}^{*}$ be half the sum of the positive roots of $\U0001d524$ and define $\stackrel{\sim}{\rho}\in {\stackrel{\sim}{\U0001d525}}^{*}$ by $\stackrel{\sim}{\rho}{|}_{\U0001d525}=\rho $ and $$\stackrel{\sim}{\rho}\left(K\right)=1+\u27e8\rho ,{\theta}^{\vee}\u27e9=\text{dual Coxeter number of}\hspace{0.17em}\U0001d524\text{,}$$ where $\theta $ (as earlier) is the highest root of $\U0001d524$ and ${\theta}^{\vee}$ is the associated coroot. Define the shifted action of $\stackrel{\sim}{W}$ on ${\stackrel{\sim}{\U0001d525}}^{*}$ by $w*\beta =w(\beta +\stackrel{\sim}{\rho})-\stackrel{\sim}{\rho},$ for $\beta \in {\stackrel{\sim}{\U0001d525}}^{*}$ and $w\in \stackrel{\sim}{W}\text{.}$ Fix a positive integer $\ell \text{.}$ Let ${P}^{+}\subset {\U0001d525}^{*}$ be the set of dominant integral weights of $\U0001d524$ and let ${P}_{\ell}^{+}\u2254\{\lambda \in {P}^{+}:\u27e8\lambda ,{\theta}^{\vee}\u27e9\le \ell \}$ be the fundamental alcove.

Define the loop algebra $\mathrm{\Omega}\left(\U0001d524\right)\u2254\U0001d524{\otimes}_{\u2102}\u2102\left[{t}^{\pm 1}\right]$ with Lie bracket $[X\otimes P,Y\otimes Q]=[X,Y]\otimes PQ,$ for $X,Y\in \U0001d524$ and $P,Q\in \u2102\left[{t}^{\pm 1}\right]\text{.}$ Then $\stackrel{\sim}{\U0001d524}$ can be viewed as a one-dimensional central extension of $\mathrm{\Omega}\left(\U0001d524\right)\text{:}$ $$\begin{array}{cc}0\u27f6\u2102K\u27f6\stackrel{\sim}{\U0001d524}\stackrel{\pi}{\u27f6}\mathrm{\Omega}\left(\U0001d524\right)\u27f60,& \text{(1.1)}\end{array}$$ where the Lie algebra homomorphism $\pi $ is defined by $\pi (X\otimes P)=X\otimes P$ and $\pi \left(K\right)=0\text{.}$

Fix an irreducible (finite-dimensional) representation $V$ of $\U0001d524$ and a number $\ell \in \u2102$
(to be called the *level* or *central charge*). Then we define the associated *generalized Verma module* for $\stackrel{\sim}{\U0001d524}$ as
$$M(V,\ell )=U\left(\stackrel{\sim}{\U0001d524}\right){\otimes}_{U\left(\stackrel{\sim}{\U0001d52d}\right)}{I}_{\ell}\left(V\right),$$
where the $\stackrel{\sim}{\U0001d52d}\text{-module}$ ${I}_{\ell}\left(V\right)$
has the same underlying vector space as $V$ on which $\stackrel{\sim}{\U0001d532}$ acts trivially, the central element
$K$ acts via the scalar $\ell $ and the action of $\U0001d524=\U0001d524\otimes {t}^{0}$
is via the $\U0001d524\text{-module}$ structure on $V\text{.}$

In the case when $\ell $ is a positive integer (in fact, it suffices to assume that $\ell \ne -h,$ where $h$ is the dual Coxeter number of $\U0001d524\text{),}$ $M(V,\ell )$ has a unique irreducible quotient, denoted $L(V,\ell )\text{.}$

It is easy to see that any vector $v\in M(V,\ell )$ is contained in a finite dimensional $\U0001d524\text{-submodule}$ of $M(V,\ell )\text{.}$ In particular, the same property holds for any vector in $L(V,\ell )\text{.}$

Consider the Lie subalgebra ${\U0001d52f}^{0}$ of $\stackrel{\sim}{\U0001d524}$ spanned by $\{{Y}_{\theta}\otimes t,{\theta}^{\vee}\otimes 1,{X}_{\theta}\otimes {t}^{-1}\},$ where ${Y}_{\theta}$ (resp. ${X}_{\theta}\text{)}$ is a non-zero root vector of $\U0001d524$ corresponding to the root $-\theta $ (resp. $\theta \text{)}$ and the coroot ${\theta}^{\vee}$ is to be thought of as an element of $\U0001d525\text{.}$ Then the Lie algebra ${\U0001d52f}^{0}$ is isomorphic to $sl\left(2\right)\text{.}$

A $\stackrel{\sim}{\U0001d524}\text{-module}$ $W$ is said to be *integrable* if every vector
$v\in W$ is contained in a finite-dimensional $\U0001d524\text{-submodule}$ of $W$
and also $v$ is contained in a finite-dimensional ${\U0001d52f}^{0}\text{-submodule}$ of $W\text{.}$

Then it follows easily from the $sl\left(2\right)\text{-theory}$ that the irreducible module $L(V,\ell )$ is integrable if and only if $\ell $ is an integer and $\ell \ge (\lambda ,{\theta}^{\vee}),$ where $\lambda $ is the highest weight of $V\text{.}$