Unipotent Hecke algebras: the structure, representation theory, and combinatorics

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

Last updated: 26 March 2015

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

Representation theory in the $G={GL}_n\left({𝔽}_q\right)$ case

This chapter examines the representation theory of unipotent Hecke algebras when $G={GL}_n\left({𝔽}_q\right)\text{.}$ The combinatorics associated with the representation theory of ${GL}_n\left({𝔽}_q\right)$ generalizes the tableaux combinatorics of the symmetric group, and one of the main results of this chapter is to also give a generalization of the RSK correspondence (see Section 2.3.4).

Fix a homomorphism $\psi :{𝔽}_q^{+}\to {ℂ}^{*}\text{.}$ Recall that for any composition $\mu \models n,$ ${\mathscr{H}}_{\mu }$ is a unipotent Hecke algebra (see Chapter 4). A fundamental result concerning Gelfand-Graev Hecke algebras is

([GGr1962],[Yok1968],[Ste1967]). For all $n>0,$ ${\mathscr{H}}_{\left(n\right)}$ is commutative.

and it will follow from Theorem 5.7, via the representation theory of unipotent Hecke algebras.

The representation theory of ${\mathscr{H}}_{\mu }$

Let $𝒮$ be a set. An $𝒮\text{-partition}$ $\lambda =({\lambda }^{\left(s_1\right)},{\lambda }^{\left(s_2\right)},\dots )$ is a sequence of partitions indexed by the elements of $𝒮\text{.}$ Let $$\begin{array}{cc}{\displaystyle {𝒫}^{𝒮}=\left\{𝒮\text{-partitions}\right\}\text{.}}& \text{(5.1)}\end{array}$$ The following discussion defines two sets $\mathrm{\Theta }$ and $\mathrm{\Phi },$ so that $\mathrm{\Theta }\text{-partitions}$ index the irreducible characters of $G$ and $\mathrm{\Phi }\text{-partitions}$ index the conjugacy classes of $G\text{.}$

Let $L_n=\text{Hom}({𝔽}_{q^n}^{*},{ℂ}^{*})$ be the character group of ${𝔽}_{q^n}^{*}\text{.}$ If $\gamma \in L_m,$ then let $$\begin{array}{crcl}{\gamma }_{\left(r\right)}:& {𝔽}_{q^{mr}}^{*}& ⟶& {ℂ}^{*}\\ & x& ⟼& \gamma \left(x^{1+q^r+q^{2r}+\cdots +q^{m(r-1)}}\right)\end{array}$$ Thus if $n=mr,$ then we may view $L_m\subseteq L_n$ by identifying $\gamma \in L_m$ with ${\gamma }_{\left(r\right)}\in L_n\text{.}$ Define $$L=\bigcup _{n\ge 0}{L}_n\text{.}$$ The Frobenius maps are $$\begin{array}{cccc}F:& {\overline{𝔽}}_q& ⟶& {\overline{𝔽}}_q\\ & x& ⟼& x^q\end{array}\text{and}\begin{array}{cccc}F:& L& ⟶& L\\ & \gamma & ⟼& {\gamma }^q\end{array},$$ where ${\overline{𝔽}}_q$ is the algebraic closure of ${𝔽}_q\text{.}$

The map $$\begin{array}{rcl}\left\{F\text{-orbits of}\hspace{0.17em}{\overline{𝔽}}_q^{*}\right\}& ⟶& \{f\in {𝔽}_q\left[t\right]\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is monic, irreducible, and}\hspace{0.17em}f\left(0\right)\ne 0\}\\ \{x,x^q,x^{q^2},\dots ,x^{q^{k-1}}\}& ⟼& f_x=\prod _{i=1}^{k-1}(t-x^{q^i}),\text{where}\hspace{0.17em}{x}^{q^k}=x\in {\overline{𝔽}}_q^{*}\end{array}$$ is a bijection such that the size of the $F\text{-orbit}$ of $x$ equals the degree $d\left(f_x\right)$ of $f_x\text{.}$ Let $$\begin{array}{cc}{\displaystyle \mathrm{\Phi }=\{f\in ℂ\left[t\right]\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is monic, irreducible and}\hspace{0.17em}f\left(0\right)\ne 0\}\text{and}\mathrm{\Theta }=\left\{F\text{-orbits in}\hspace{0.17em}L\right\}\text{.}}& \text{(5.2)}\end{array}$$ If $\eta$ is a $\mathrm{\Phi }\text{-partition}$ and $\lambda$ is a $\mathrm{\Theta }\text{-partition,}$ then let $$\mid \eta \mid =\sum _{f\in \mathrm{\Phi }}d\left(f\right)\mid {\eta }^{\left(f\right)}\mid \text{and}\mid \lambda \mid =\sum _{\phi \in \mathrm{\Theta }}\mid \phi \mid \mid {\lambda }^{\left(\phi \right)}\mid$$ be the size of $\eta$ and $\lambda ,$ respectively. Let the sets ${𝒫}^{\mathrm{\Phi }}$ and ${𝒫}^{\mathrm{\Theta }}$ be as in (5.1) and let $$\begin{array}{cc}{\displaystyle {𝒫}_n^{\mathrm{\Phi }}=\{\eta \in {𝒫}^{\mathrm{\Phi }}\hspace{0.17em}|\hspace{0.17em}\mid \eta \mid =n\}\text{and}{𝒫}_n^{\mathrm{\Theta }}=\{\lambda \in {𝒫}^{\mathrm{\Theta }}\hspace{0.17em}|\hspace{0.17em}\mid \lambda \mid =n\}\text{.}}& \text{(5.3)}\end{array}$$

(Green [Gre1955]). Let $G_n={GL}_n\left({𝔽}_q\right)\text{.}$ (a) ${𝒫}_n^{\mathrm{\Phi }}$ indexes the conjugacy classes $K^{\eta }$ of $G_n,$ (b) ${𝒫}_n^{\mathrm{\Theta }}$ indexes the irreducible $G_n\text{-modules}$ $G_n^{\lambda }\text{.}$

Suppose $\lambda \in {𝒫}^{\mathrm{\Theta }}\text{.}$ A column strict tableau $P=(P^{\left({\phi }_1\right)},P^{\left({\phi }_2\right)},\dots )$ of shape $\lambda$ is a column strict filling of $\lambda$ by positive integers. That is, $P^{\left(\phi \right)}$ is a column strict tableau of shape ${\lambda }^{\left(\phi \right)}\text{.}$ Write $\text{sh}\left(P\right)=\lambda \text{.}$ The weight of $P$ is the composition $\text{wt}\left(P\right)=(\text{wt}{\left(P\right)}_1,\text{wt}{\left(P\right)}_2,\dots )$ given by $$\text{wt}{\left(P\right)}_i=\sum _{\phi \in \mathrm{\Theta }}\mid \phi \mid \left(\text{numbers of}\hspace{0.17em}i\hspace{0.17em}\text{in}\hspace{0.17em}{P}^{\left(\phi \right)}\right)\text{.}$$

If $\lambda \in {𝒫}^{\mathrm{\Theta }}$ and $\mu$ is a composition, then let $$\begin{array}{cc}{\displaystyle {\hat{\mathscr{H}}}_{\mu }^{\lambda }=\left\{\text{column strict tableaux}\hspace{0.17em}P\hspace{0.17em}\right|\hspace{0.17em}\text{sh}\left(P\right)=\lambda \text{, wt}\left(P\right)=\mu \}}& \text{(5.4)}\end{array}$$ and $$\begin{array}{cc}{\displaystyle {\hat{\mathscr{H}}}_{\mu }=\{\lambda \in {𝒫}^{\mathrm{\Theta }}\hspace{0.17em}|\hspace{0.17em}{\hat{\mathscr{H}}}_{\mu }^{\lambda }\hspace{0.17em}\text{is not empty}\}\text{.}}& \text{(5.5)}\end{array}$$

The following theorem is a consequence of Theorem 2.3 and a theorem proved by Zelevinsky [Zel1981-2] (see Theorem 5.5). A proof of Zelevinsky’s theorem is in Section 5.3.

The set ${\hat{\mathscr{H}}}_{\mu }$ indexes the irreducible ${\mathscr{H}}_{\mu }\text{-modules}$ ${\mathscr{H}}_{\mu }^{\lambda }$ and $$\text{dim}\left({\mathscr{H}}_{\mu }^{\lambda }\right)=\mid {\hat{\mathscr{H}}}_{\mu }^{\lambda }\mid \text{.}$$

A generalization of the RSK correspondence

For a composition $\mu \models n,$ let $N_{\mu }$ be as in (4.11) and $M_{\mu }$ as in (4.12).

The $({\mathscr{H}}_{\mu },{\mathscr{H}}_{\mu })\text{-bimodule}$ decomposition $${\mathscr{H}}_{\mu }\cong \underset{\lambda \in {\hat{\mathscr{H}}}_{\mu }}{⨁}{\mathscr{H}}_{\mu }^{\lambda }\otimes {\mathscr{H}}_{\mu }^{\lambda }\text{implies}\mid N_{\mu }\mid =\text{dim}\left({\mathscr{H}}_{\mu }\right)=\sum _{\lambda \in {\hat{\mathscr{H}}}_{\mu }}\text{dim}{\left({\mathscr{H}}_{\mu }^{\lambda }\right)}^2=\sum _{\lambda \in {\hat{\mathscr{H}}}_{\mu }}{\mid {\hat{\mathscr{H}}}_{\mu }^{\lambda }\mid }^2\text{.}$$ Theorem 5.4, below, gives a combinatorial proof of this identity.

Encode each matrix $a\in M_{\mu }$ as a $\mathrm{\Phi }\text{-sequence}$ $$(a^{\left(f_1\right)},a^{\left(f_2\right)},\dots ),f_i\in \mathrm{\Phi },$$ where $a^{\left(f\right)}\in M_{\ell \left(\mu \right)}\left({\mathbb{Z}}_{\ge 0}\right)$ is given by $$a_{ij}^{\left(f\right)}=\text{highest power of}\hspace{0.17em}f\hspace{0.17em}\text{dividing}\hspace{0.17em}{a}_{ij}\text{.}$$ Note that this is an entry by entry “factorization” of $a$ such that $$a_{ij}=\prod _{f\in \mathrm{\Phi }}{f}^{a_{ij}^{\left(f\right)}}\text{.}$$ Recall from Section 2.3.4 the classical RSK correspondence $$\begin{array}{rcl}{M}_{\ell }\left({\mathbb{Z}}_{\ge 0}\right)& ⟶& \left\{\text{Pairs}\hspace{0.17em}(P,Q)\hspace{0.17em}\text{of column strict tableaux of the same shape}\right\}\\ b& ⟼& (P\left(b\right),Q\left(b\right))\text{.}\end{array}$$

For $a\in M_{\mu },$ let $P\left(a\right)$ and $Q\left(a\right)$ be the $\mathrm{\Phi }\text{-column}$ strict tableaux given by $$P\left(a\right)=(P\left(a^{\left(f_1\right)}\right),P\left(a^{\left(f_2\right)}\right),\dots )\text{and}Q\left(a\right)=(Q\left(a^{\left(f_1\right)}\right),Q\left(a^{\left(f_2\right)}\right),\dots )\text{for}\hspace{0.17em}{f}_i\in \mathrm{\Phi }\text{.}$$ Then the map $$\begin{array}{rcccl}{N}_{\mu }& ⟶& M_{\mu }& ⟶& \left\{\begin{array}{c}\text{Pairs}\hspace{0.17em}(P,Q)\hspace{0.17em}\text{of}\hspace{0.17em}\mathrm{\Phi }\text{-column}\\ \text{strict tableaux of the}\\ \text{same shape and weight of}\hspace{0.17em}\mu \end{array}\right\}\\ v& ⟼& a_v& ⟼& (P\left(a_v\right),Q\left(a_v\right)),\end{array}$$ is a bijection, where the first map is the inverse of the bijection in Theorem 4.2.

By the construction above, the map is well-defined, and since all the steps are invertible, the map is a bijection.

Example. Suppose $\mu =(7,5,3,2)$ and $f,g,h\in \mathrm{\Phi }$ are such that $d\left(f\right)=1,$ $d\left(g\right)=2,$ and $d\left(h\right)=3\text{.}$ Then $$a_v=\left(\begin{array}{cccc}g& f^{2}h& 1& 1\\ h& 1& g& 1\\ 1& 1& f& f^2\\ g& 1& 1& 1\end{array}\right)\in M_{\begin{array}{c} \end{array}}$$ corresponds to the sequence $$(a_v^{\left(f_1\right)},a_v^{\left(f_2\right)},\dots )=({\left(\begin{array}{cccc}0& 2& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 2\\ 0& 0& 0& 0\end{array}\right)}^{\left(f\right)},{\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right)}^{\left(g\right)},{\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)}^{\left(h\right)})$$ and $$(P\left(a_v\right),Q\left(a_v\right))=({\begin{array}{c} 2 2 4 3 4 \end{array}}^{\left(f\right)},{\begin{array}{c} 1 1 3 \end{array}}^{\left(g\right)},{\begin{array}{c} 1 2 \end{array}}^{\left(h\right)})({\begin{array}{c} 1 1 3 3 3 \end{array}}^{\left(f\right)},{\begin{array}{c} 1 4 2 \end{array}}^{\left(g\right)},{\begin{array}{c} 1 2 \end{array}}^{\left(h\right)})\text{.}$$

Zelevinsky’s decomposition of ${\text{Ind}}_U^G\left({\psi }_{\mu }\right)$

This section proves the theorem

(Zelevinsky [Zel1981-2]). Let $U$ be the subgroup of unipotent upper-triangular matrices of $G={GL}_n\left({𝔽}_q\right),$ $\mu \models n$ and ${\psi }_{\mu }$ be as in (4.8). Then $${\text{Ind}}_U^G\left({\psi }_{\mu }\right)=\underset{\lambda \in {\hat{\mathscr{H}}}_{\mu }}{⨁}\text{Card}\left({\hat{\mathscr{H}}}_{\mu }^{\lambda }\right)G^{\lambda }\text{.}$$

Theorem 5.3 follows from this theorem and Theorem 2.3. The proof of Theorem 5.5 is in 3 steps. (1) Establish the necessary connection between symmetric functions and the representation theory of $G\text{.}$ (2) Prove Theorem 5.5 for the case when $\ell \left(\mu \right)=1\text{.}$ (3) Generalize (2) to arbitrary $\mu \text{.}$ The proof below uses the ideas of Zelevinsky’s proof, but explicitly uses symmetric functions to prove the results. Specifically, the following discussion through the proof of Theorem 5.7 corresponds to [Zel1981-2, Sections 9-11] and Theorem 5.5 corresponds to [Zel1981-2, Theorem 12.1].

Preliminaries to the proof (1)

Let $\mu \models n$ and $G=G_n\text{.}$ The group $$\begin{array}{cc}{\displaystyle P_{\mu }=\left\{(\begin{array}{cccc}\begin{array}{|c|}\hline g_1\\ \hline\end{array}& & & *\\ & \begin{array}{|c|}\hline g_2\\ \hline\end{array}& & \\ & & \ddots & \\ 0& & & \begin{array}{|c|}\hline g_{\ell }\\ \hline\end{array})\end{array}\hspace{0.17em}\right|\hspace{0.17em}{g}_i\in G_{{\mu }_i}={GL}_{{\mu }_i}\left({𝔽}_q\right)\}}& \text{(5.6)}\end{array}$$ has subgroups $$\begin{array}{cc}{\displaystyle L_{\mu }=G_{{\mu }_1}\oplus G_{{\mu }_2}\oplus \cdots \oplus G_{{\mu }_{\ell }}\text{and}{U}_{\mu }=\left\{(\begin{array}{cccc}\begin{array}{|c|}\hline {\text{Id}}_{{\mu }_1}\\ \hline\end{array}& & & *\\ & \begin{array}{|c|}\hline {\text{Id}}_{{\mu }_2}\\ \hline\end{array}& & \\ & & \ddots & \\ 0& & & \begin{array}{|c|}\hline {\text{Id}}_{{\mu }_{\ell }}\\ \hline\end{array})\end{array}\right\},}& \text{(5.7)}\end{array}$$ where ${\text{Id}}_k$ is the $k\times k$ identity matrix. Note that $P_{\mu }=L_{\mu }{U}_{\mu }$ and $P_{\mu }=N_G\left(U_{\mu }\right)\text{.}$ The indflation map is a composition of the inflation map and the induction map, $$\begin{array}{cccccc}{\text{Indf}}_{L_{\mu }}^G:& R\left[L_{\mu }\right]& ⟶& R\left[P_{\mu }\right]& ⟶& R\left[G\right]\\ & \chi & ⟼& {\text{Inf}}_{L_{\mu }}^{P_{\mu }}\left(\chi \right)& ⟼& {\text{Ind}}_{P_{\mu }}^G\left({\text{Inf}}_{L_{\mu }}^{P_{\mu }}\left(\chi \right)\right),\end{array}$$ where $$\begin{array}{cccc}{\text{Inf}}_{L_{\mu }}^{P_{\mu }}\left(\chi \right):& P_{\mu }& ⟶& ℂ\\ & lu& ⟼& \chi \left(l\right),\end{array}\text{for}\hspace{0.17em}l\in L_{\mu }\hspace{0.17em}\text{and}\hspace{0.17em}u\in U_{\mu }\text{.}$$

Suppose $\lambda \in {𝒫}^{\mathrm{\Theta }}$ and $\eta \in {𝒫}^{\mathrm{\Phi }}$ (see (5.3)). Let ${\chi }^{\lambda }$ be the irreducible character corresponding to the irreducible $G\text{-module}$ $G^{\lambda }$ and let ${\kappa }^{\eta }$ be the characteristic function corresponding to the conjugacy class $K^{\eta }$ (see Theorem 5.2), given by $${\kappa }^{\eta }\left(g\right)=\{\begin{array}{ll}1,& \text{if}\hspace{0.17em}g\in K^{\eta },\\ 0,& \text{otherwise,}\end{array}\text{for}\hspace{0.17em}g\in G_{\mid \eta \mid }\text{.}$$ Define $$R=\underset{n\ge 0}{⨁}R\left[G_n\right]=ℂ\text{-span}\left\{{\chi }^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {𝒫}^{\mathrm{\Theta }}\}=ℂ\text{-span}\left\{{\kappa }^{\eta }\hspace{0.17em}\right|\hspace{0.17em}\eta \in {𝒫}^{\mathrm{\Phi }}\}\text{.}$$ The space $R$ has an inner product defined by $$⟨{\chi }^{\lambda },{\chi }^{\nu }⟩={\delta }_{\lambda \nu },$$ and multiplication $$\begin{array}{cc}{\displaystyle {\chi }^{\lambda }\circ {\chi }^{\nu }={\text{Indf}}_{L_{(r,s)}}^{G_{r+s}}({\chi }^{\lambda }\otimes {\chi }^{\nu }),\text{for}\hspace{0.17em}\lambda \in {𝒫}_r^{\mathrm{\Theta }},\nu \in {𝒫}_s^{\mathrm{\Theta }}\text{.}}& \text{(5.8)}\end{array}$$

For each $\phi \mathrm{\Theta },$ let $\{Y_1^{\left(\phi \right)},Y_2^{\left(\phi \right)},\dots \}$ be an infinite set of variables, and let $${\mathrm{\Lambda }}_{ℂ}=\underset{\phi \in \mathrm{\Theta }}{⨂}{\mathrm{\Lambda }}_{ℂ}\left(Y^{\left(\phi \right)}\right),$$ where ${\mathrm{\Lambda }}_{ℂ}\left(Y^{\left(\phi \right)}\right)$ is the ring of symmetric functions in $\{Y_1^{\left(\phi \right)},Y_2^{\left(\phi \right)},\dots \}$ (see Section 2.3.3). For each $f\in \mathrm{\Phi },$ define an additional set of variables $\{X_1^{\left(f\right)},X_2^{\left(f\right)},\dots \}$ such that the symmetric functions in the $Y$ variables are related to the symmetric functions in the $X$ variables by the transform $$\begin{array}{cc}{\displaystyle p_k\left(Y^{\left(\phi \right)}\right)={(-1)}^{k\mid \phi \mid -1}\sum _{x\in {𝔽}_{q^{k\mid \phi \mid }}^{*}}\xi \left(x\right)p_{\frac{k\mid \phi \mid }{d\left(f_x\right)}}\left(X^{\left(f_x\right)}\right),}& \text{(5.9)}\end{array}$$ where $\xi \in \phi ,$ $f_x\in \mathrm{\Phi }$ is the irreducible polynomial that has $x$ as a root, and $p_{\frac{a}{b}}\left(X^{\left(f\right)}\right)=0$ if $\frac{a}{b}\notin {\mathbb{Z}}_{\ge 0}\text{.}$ Then $${\mathrm{\Lambda }}_{ℂ}=\underset{f\in \mathrm{\Phi }}{⨂}{\mathrm{\Lambda }}_{ℂ}\left(X^{\left(f\right)}\right)\text{.}$$

For $\nu \in 𝒫,$ let $s_{\nu }\left(Y^{\left(\phi \right)}\right)$ be the Schur function and $P_{\nu }(X^{\left(f\right)};t)$ be the Hall-Littlewood symmetric function (as in Section 2.3.3). Define $$\begin{array}{cc}{\displaystyle s_{\lambda }=\prod _{\phi \in \mathrm{\Theta }}{s}_{{\lambda }^{\left(\phi \right)}}\left(Y^{\left(\phi \right)}\right)\text{and}{P}_{\eta }=q^{-n\left(\eta \right)}\prod _{f\in \mathrm{\Phi }}{P}_{{\eta }^{\left(f\right)}}(X^{\left(f\right)};q^{-d\left(f\right)}),}& \text{(5.10)}\end{array}$$ where $n\left(\eta \right)=\sum _{f\in \mathrm{\Theta }}d\left(f\right)n\left({\eta }^{\left(f\right)}\right)$ and for a composition $\mu ,$ $n\left(\mu \right)=\sum _{i=1}^{\ell \left(\mu \right)}(i-1){\mu }_i\text{.}$ The ring $${\mathrm{\Lambda }}_{ℂ}=ℂ\text{-span}\left\{s_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {𝒫}^{\mathrm{\Theta }}\}-ℂ\text{-span}\left\{P_{\eta }\hspace{0.17em}\right|\hspace{0.17em}\eta \in {𝒫}^{\mathrm{\Phi }}\}$$ has an inner product given by $$⟨s_{\lambda },s_{\nu }⟩={\delta }_{\lambda \nu }\text{.}$$

(Green [Gre1955],Macdonald [Mac1995]). The linear map $$\begin{array}{cccl}\text{ch}:& R& ⟶& {\mathrm{\Lambda }}_{ℂ}\\ & {\chi }^{\lambda }& ⟼& s_{\lambda },\text{for}\hspace{0.17em}\lambda \in {𝒫}^{\mathrm{\Theta }}\\ & {\kappa }^{\eta }& ⟼& P_{\eta },\text{for}\hspace{0.17em}\eta \in {𝒫}^{\mathrm{\Phi }},\end{array}$$ is an algebra isomorphism that preserves the inner product.

A unipotent conjugacy class $K^{\eta }$ is a conjugacy class such that ${\eta }^{\left(f\right)}=\varnothing$ unless $f=t-1\text{.}$ Let $$𝒰=ℂ\text{-span}\left\{{\kappa }^{\eta }\hspace{0.17em}\right|\hspace{0.17em}{\eta }^{\left(f\right)}=\varnothing \text{, unless}\hspace{0.17em}f=t-1\}\subseteq R$$ be the subalgebra of unipotent class functions. Note that by (5.10) and Theorem 5.6, $$\text{ch}\left(𝒰\right)={\mathrm{\Lambda }}_{ℂ}\left(X^{(t-1)}\right)\text{.}$$ Consider the projection $\pi :R\to 𝒰$ which is an algebra homomorphism given by $$\left(\pi {\chi }^{\lambda }\right)\left(g\right)=\{\begin{array}{ll}{\chi }^{\lambda }\left(g\right),& \text{if}\hspace{0.17em}g\in G\hspace{0.17em}\text{is unipotent,}\\ 0,& \text{otherwise,}\end{array}\lambda \in {𝒫}^{\mathrm{\Theta }}\text{.}$$ Then $\stackrel{\sim }{\pi }=\text{ch}\circ \pi \circ {\text{ch}}^{-1}:{\mathrm{\Lambda }}_{ℂ}\to {\mathrm{\Lambda }}_{ℂ}\left(X^{(t-1)}\right)$ is given by $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \stackrel{\sim }{\pi }\left(p_k\left(Y^{\left(\phi \right)}\right)\right)}& =& {\displaystyle \stackrel{\sim }{\pi }\left({(-1)}^{k\mid \phi \mid -1}\sum _{x\in {𝔽}_{q^{k\mid \phi \mid }}^{*}}\xi \left(x\right)p_{\frac{k\mid \phi \mid }{d\left(f_x\right)}}\left(X^{\left(f_x\right)}\right)\right)\text{(by (5.9))}}\\ & =& {\displaystyle {(-1)}^{k\mid \phi \mid -1}\xi \left(1\right)p_{\frac{k\mid \phi \mid }{1}}\left(X^{(t-1)}\right)+0}\\ & =& {\displaystyle {(-1)}^{k\mid \phi \mid -1}{p}_{k\mid \phi \mid }\left(X^{(t-1)}\right)\text{.}}\end{array}}& \text{(5.11)}\end{array}$$

The decomposition of ${\text{Ind}}_U^G\left({\psi }_{\left(n\right)}\right)$ (2)

The representation ${\text{Ind}}_U^G\left({\psi }_{\left(n\right)}\right)$ is the Gelfand-Graev module, and with Theorem 2.3, Theorem 5.7, below, proves that ${\mathscr{H}}_{\left(n\right)}$ is commutative.

Let $U$ be the subgroup of unipotent upper-triangular matrices of $G={GL}_n\left({𝔽}_q\right)\text{.}$ Then $$\text{ch}\left({\text{Ind}}_U^G\left({\psi }_{\left(n\right)}\right)\right)=\sum _{\underset{\text{ht}\left(\lambda \right)=1}{\lambda \in {𝒫}_n^{\mathrm{\Theta }}}}{s}_{\lambda },\text{where}\hspace{0.17em}\text{ht}\left(\lambda \right)=\text{max}\left\{\ell \left({\lambda }^{\left(\phi \right)}\right)\hspace{0.17em}\right|\hspace{0.17em}\phi \in \mathrm{\Theta }\}\text{.}$$

Decmoposition of ${\text{Ind}}_U^G\left({\psi }_{\mu }\right)$ (3)

Suppose $\lambda ,\nu \in {𝒫}^{\mathrm{\Theta }}\text{.}$ A column strict tableau $P$ of shape $\lambda$ and weight $\nu$ is a column strict filling of $\lambda$ such that for each $\phi \in \mathrm{\Theta },$ $$\text{sh}\left(P^{\left(\phi \right)}\right)={\lambda }^{\left(\phi \right)}\text{and}\text{wt}\left(P^{\left(\phi \right)}\right)={\nu }^{\left(\phi \right)}\text{.}$$ We can now prove the theorem stated at the beginning of this section:

([Zel1981-2]) Let $U$ be the subgroup of unipotent upper-triangular matrices of $G={GL}_n\left({𝔽}_q\right),$ $\mu \models n$ and ${\psi }_{\mu }$ be as in (4.8). Then $${\text{Ind}}_U^G\left({\psi }_{\mu }\right)=\underset{\lambda \in {\hat{\mathscr{H}}}_{\mu }}{⨁}\text{Card}\left({\hat{\mathscr{H}}}_{\mu }^{\lambda }\right)G^{\lambda }\text{.}$$

A weight space decomposition of ${\mathscr{H}}_{\mu }\text{-modules}$

Let $\mu =({\mu }_1,{\mu }_2,\dots ,{\mu }_{\ell })\models n$ and let $P_{\mu },$ $L_{\mu }$ and $U_{\mu }$ be as in (5.6) and (5.7). Recall that $$e_{\mu }=\frac{1}{\mid U\mid }\sum _{u\in U}{\psi }_{\mu }\left(u^{-1}\right)u\text{.}$$

For $a\in M_{\mu },$ let $T_a=e_{\mu }{v}_a{e}_{\mu }$ with $v_a$ as in (4.14). Then the map $$\begin{array}{rcl}{\mathscr{H}}_{\left({\mu }_1\right)}\otimes {\mathscr{H}}_{\left({\mu }_2\right)}\otimes \cdots \otimes {\mathscr{H}}_{\left({\mu }_{\ell }\right)}& ⟶& {\mathscr{H}}_{\mu }\\ T_{\left(f_1\right)}\otimes T_{\left(f_2\right)}\otimes \cdots \otimes T_{\left(f_{\ell }\right)}& ⟼& T_{\left(f_1\right)\oplus \left(f_2\right)\oplus \cdots \oplus \left(f_{\ell }\right)},& \text{for}\hspace{0.17em}\left(f_i\right)\in M_{\left({\mu }_i\right)}\end{array}$$ is an injective algebra homomorphism with image ${ℒ}_{\mu }=e_{\mu }{P}_{\mu }{e}_{\mu }=e_{\mu }{L}_{\mu }{e}_{\mu }\text{.}$

Remark. This is the ${GL}_n\left({𝔽}_q\right)$ version of Corollary 3.5.

Let ${ℒ}_{\mu }$ be as in Theorem 5.8. By Theorem 5.1 each ${\mathscr{H}}_{\left({\mu }_i\right)}$ is commutative, so ${ℒ}_{\mu }$ is commutative and all the irreducible ${ℒ}_{\mu }\text{-modules}$ ${ℒ}_{\mu }^{\gamma }$ are one-dimensional. Theorem 5.3 implies that $${\hat{\mathscr{H}}}_{\left({\mu }_i\right)}=\left\{\mathrm{\Theta }\text{-partitions}\hspace{0.17em}\lambda \hspace{0.17em}\right|\hspace{0.17em}\mid \lambda \mid ={\mu }_i,\text{ht}\left(\lambda \right)=1\}$$ indexes the irreducible ${\mathscr{H}}_{\left({\mu }_i\right)}\text{-modules.}$ Therefore, the set $$\begin{array}{cc}{\displaystyle {\widehat{ℒ}}_{\mu }={\hat{\mathscr{H}}}_{\left({\mu }_1\right)}\times {\hat{\mathscr{H}}}_{\left({\mu }_2\right)}\times \cdots \times {\hat{\mathscr{H}}}_{\left({\mu }_{\ell }\right)}=\{\gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_{\ell })\hspace{0.17em}|\hspace{0.17em}{\gamma }_i\in {\hat{\mathscr{H}}}_{\left({\mu }_i\right)}\}}& \text{(5.14)}\end{array}$$ indexes the irreducible ${ℒ}_{\mu }\text{-modules.}$ Identify $\gamma \in {\widehat{ℒ}}_{\mu }$ with the map $\gamma :{ℒ}_{\mu }\to ℂ$ such that $$yv=\gamma \left(y\right)v,\text{for all}\hspace{0.17em}y\in {ℒ}_{\mu },v\in {ℒ}_{\mu }^{\gamma }\text{.}$$

For $\gamma \in {\widehat{ℒ}}_{\mu },$ the $\gamma \text{-weight}$ space $V_{\gamma }$ of an ${\mathscr{H}}_{\mu }\text{-module}$ $V$ is $$V_{\gamma }=\{v\in V\hspace{0.17em}|\hspace{0.17em}yv=\gamma \left(y\right)v\text{, for all}\hspace{0.17em}y\in {ℒ}_{\mu }\}\text{.}$$ Then $$V\cong \underset{\gamma \in {\widehat{ℒ}}_{\mu }}{⨁}{V}_{\gamma }\text{.}$$

Let $\lambda \in {𝒫}^{\mathrm{\Theta }}$ and $\gamma \in {\widehat{ℒ}}_{\mu }\text{.}$ A column strict tableau $P$ of shape $\lambda$ and weight $\gamma$ is column strict filling of $\lambda$ such that for each $\phi \in \mathrm{\Theta },$ $$\text{sh}\left(P^{\left(\phi \right)}\right)={\lambda }^{\left(\phi \right)}\text{and}\text{wt}\left(P^{\left(\phi \right)}\right)=(\mid {\gamma }_1^{\left(\phi \right)}\mid ,\mid {\gamma }_2^{\left(\phi \right)}\mid ,\dots ,\mid {\gamma }_{\ell }^{\left(\phi \right)}\mid ),$$ where $\mid {\gamma }_i^{\left(\phi \right)}\mid$ is the number of boxes in the partition ${\gamma }_i^{\left(\phi \right)}$ (which has length 1). Let $${\hat{\mathscr{H}}}_{\gamma }^{\lambda }=\{P\in {\mathscr{H}}_{\mu }^{\lambda }\hspace{0.17em}|\hspace{0.17em}\text{sh}\left(P\right)=\lambda \text{, wt}\left(P\right)=\gamma \}\text{.}$$

For example, suppose $$\lambda =({\begin{array}{c} \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} \end{array}}^{\left({\phi }_3\right)})$$ and $$\gamma =({\begin{array}{c}\end{array}}^{\left({\phi }_1\right)},{\begin{array}{c}\end{array}}^{\left({\phi }_2\right)},{\begin{array}{c}\end{array}}^{\left({\phi }_3\right)})\otimes ({\begin{array}{c}\end{array}}^{\left({\phi }_1\right)},{\begin{array}{c}\end{array}}^{\left({\phi }_3\right)})\otimes ({\begin{array}{c}\end{array}}^{\left({\phi }_1\right)},{\begin{array}{c}\end{array}}^{\left({\phi }_2\right)})\otimes \left({\begin{array}{c}\end{array}}^{\left({\phi }_2\right)}\right)\text{.}$$ Then $${\hat{\mathscr{H}}}_{\gamma }^{\lambda }=\left\{\begin{array}{cc}({\begin{array}{c} 1 1 3 2 2 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 3 4 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 2 \end{array}}^{\left({\phi }_3\right)}),& ({\begin{array}{c} 1 1 3 2 2 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 4 3 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 2 \end{array}}^{\left({\phi }_3\right)})\\ ({\begin{array}{c} 1 1 2 2 3 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 3 4 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 2 \end{array}}^{\left({\phi }_3\right)}),& ({\begin{array}{c} 1 1 2 2 3 \end{array}}^{\left({\phi }_1\right)},{\begin{array}{c} 1 4 3 \end{array}}^{\left({\phi }_2\right)},{\begin{array}{c} 1 2 \end{array}}^{\left({\phi }_3\right)})\end{array}\right\}\text{.}$$

Let ${\mathscr{H}}_{\mu }^{\lambda }$ be an irreducible ${\mathscr{H}}_{\mu }\text{-module}$ and $\gamma \in {\widehat{ℒ}}_{\mu }\text{.}$ Then $$\text{dim}{\left({\mathscr{H}}_{\mu }^{\lambda }\right)}_{\gamma }=\text{Card}\left({\hat{\mathscr{H}}}_{\gamma }^{\lambda }\right)\text{.}$$

Notes and References

This is an excerpt of the PhD thesis Unipotent Hecke algebras: the structure, representation theory, and combinatorics by F. Nathaniel Edgar Thiem.

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