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

Last updated: 26 March 2015

## 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 $\mu \models n,$ ${ℋ}_{\mu }$ is a unipotent Hecke algebra (see Chapter 4). A fundamental result concerning Gelfand-Graev Hecke algebras is

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

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

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

Let $𝒮$ be a set. An $𝒮\text{-partition}$ $\lambda =\left({\lambda }^{\left({s}_{1}\right)},{\lambda }^{\left({s}_{2}\right)},\dots \right)$ is a sequence of partitions indexed by the elements of $𝒮\text{.}$ Let $𝒫𝒮= {𝒮-partitions}. (5.1)$ 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 ${L}_{n}=\text{Hom}\left({𝔽}_{{q}^{n}}^{*},{ℂ}^{*}\right)$ be the character group of ${𝔽}_{{q}^{n}}^{*}\text{.}$ If $\gamma \in {L}_{m},$ then let $γ(r): 𝔽qmr* ⟶ ℂ* x ⟼ γ(x1+qr+q2r+⋯+qm(r-1))$ 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 $L=⋃n≥0Ln.$ The Frobenius maps are $F: 𝔽‾q ⟶ 𝔽‾q x ⟼ xq and F: L ⟶ L γ ⟼ γq ,$ where ${\stackrel{‾}{𝔽}}_{q}$ is the algebraic closure of ${𝔽}_{q}\text{.}$

The map ${F-orbits of 𝔽‾q*} ⟶ {f∈𝔽q[t] | f is monic, irreducible, and f(0)≠0} {x,xq,xq2,…,xqk-1} ⟼ fx=∏i=1k-1(t-xqi),where xqk=x∈𝔽‾q*$ 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 $Φ={f∈ℂ[t] | f is monic, irreducible and f(0)≠0} andΘ={F-orbits in L}. (5.2)$ If $\eta$ is a $\mathrm{\Phi }\text{-partition}$ and $\lambda$ is a $\mathrm{\Theta }\text{-partition,}$ then let $∣η∣=∑f∈Φ d(f)∣η(f)∣ and∣λ∣= ∑φ∈Θ∣φ∣ ∣λ(φ)∣$ 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 $𝒫nΦ={η∈𝒫Φ | ∣η∣=n} and𝒫nΘ= {λ∈𝒫Θ | ∣λ∣=n}. (5.3)$

(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 $P=\left({P}^{\left({\phi }_{1}\right)},{P}^{\left({\phi }_{2}\right)},\dots \right)$ 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 $\text{wt}\left(P\right)=\left(\text{wt}{\left(P\right)}_{1},\text{wt}{\left(P\right)}_{2},\dots \right)$ given by $wt(P)i= ∑φ∈Θ∣φ∣ (numbers of i in P(φ)).$

If $\lambda \in {𝒫}^{\mathrm{\Theta }}$ and $\mu$ is a composition, then let $ℋˆμλ= {column strict tableaux P | sh(P)=λ, wt(P)=μ} (5.4)$ and $ℋˆμ= {λ∈𝒫Θ | ℋˆμλ is not empty}. (5.5)$

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 ${\stackrel{ˆ}{ℋ}}_{\mu }$ indexes the irreducible ${ℋ}_{\mu }\text{-modules}$ ${ℋ}_{\mu }^{\lambda }$ and $dim(ℋμλ)= ∣ℋˆμλ∣.$

### 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 $\left({ℋ}_{\mu },{ℋ}_{\mu }\right)\text{-bimodule}$ decomposition $ℋμ≅⨁λ∈ℋˆμ ℋμλ⊗ℋμλ implies∣Nμ∣=dim (ℋμ)=∑λ∈ℋˆμ dim(ℋμλ)2= ∑λ∈ℋˆμ ∣ℋˆμλ∣2.$ Theorem 5.4, below, gives a combinatorial proof of this identity.

Encode each matrix $a\in {M}_{\mu }$ as a $\mathrm{\Phi }\text{-sequence}$ $(a(f1),a(f2),…), fi∈Φ,$ where ${a}^{\left(f\right)}\in {M}_{\ell \left(\mu \right)}\left({ℤ}_{\ge 0}\right)$ is given by $aij(f)= highest power of f dividing aij.$ Note that this is an entry by entry “factorization” of $a$ such that $aij=∏f∈Φ faij(f).$ Recall from Section 2.3.4 the classical RSK correspondence $Mℓ(ℤ≥0) ⟶ {Pairs (P,Q) of column strict tableaux of the same shape} b ⟼ (P(b),Q(b)).$

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 $P(a)=(P(a(f1)),P(a(f2)),…) andQ(a)= (Q(a(f1)),Q(a(f2)),…) for fi∈Φ.$ Then the map $Nμ ⟶ Mμ ⟶ { Pairs (P,Q) of Φ-column strict tableaux of the same shape and weight of μ } v ⟼ av ⟼ (P(av),Q(av)),$ 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 $\mu =\left(7,5,3,2\right)$ 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 $av= ( gf2h11 h1g1 11ff2 g111 ) ∈ M$ corresponds to the sequence $(av(f1),av(f2),…)= ( ( 0200 0000 0012 0000 ) (f) , ( 1000 0010 0000 1000 ) (g) , ( 0100 1000 0000 0000 ) (h) )$ and $(P(av),Q(av))= ( 2 2 4 3 4 (f) , 1 1 3 (g) , 1 2 (h) ) ( 1 1 3 3 3 (f) , 1 4 2 (g) , 1 2 (h) ) .$

### 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 $IndUG(ψμ)= ⨁λ∈ℋˆμ Card(ℋˆμλ) Gλ.$

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 $Pμ= { ( g1* g2 ⋱ 0gℓ ) | gi∈Gμi =GLμi(𝔽q) } (5.6)$ has subgroups $Lμ=Gμ1⊕ Gμ2⊕⋯⊕ Gμℓand Uμ= { ( Idμ1* Idμ2 ⋱ 0Idμℓ ) } , (5.7)$ where ${\text{Id}}_{k}$ is the $k×k$ 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, $IndfLμG: R[Lμ] ⟶ R[Pμ] ⟶ R[G] χ ⟼ InfLμPμ(χ) ⟼ IndPμG(InfLμPμ(χ)),$ where $InfLμPμ(χ): Pμ ⟶ ℂ lu ⟼ χ(l), for l∈Lμ and u∈Uμ.$

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 $κη(g)= { 1, if g∈Kη, 0, otherwise, for g∈G∣η∣.$ Define $R=⨁n≥0R[Gn] =ℂ-span{χλ | λ∈𝒫Θ} =ℂ-span{κη | η∈𝒫Φ}.$ The space $R$ has an inner product defined by $⟨χλ,χν⟩ =δλν,$ and multiplication $χλ∘χν= IndfL(r,s)Gr+s (χλ⊗χν), for λ∈𝒫rΘ, ν∈𝒫sΘ. (5.8)$

For each $\phi \mathrm{\Theta },$ let $\left\{{Y}_{1}^{\left(\phi \right)},{Y}_{2}^{\left(\phi \right)},\dots \right\}$ be an infinite set of variables, and let $Λℂ= ⨂φ∈Θ Λℂ(Y(φ)),$ where ${\mathrm{\Lambda }}_{ℂ}\left({Y}^{\left(\phi \right)}\right)$ is the ring of symmetric functions in $\left\{{Y}_{1}^{\left(\phi \right)},{Y}_{2}^{\left(\phi \right)},\dots \right\}$ (see Section 2.3.3). For each $f\in \mathrm{\Phi },$ define an additional set of variables $\left\{{X}_{1}^{\left(f\right)},{X}_{2}^{\left(f\right)},\dots \right\}$ such that the symmetric functions in the $Y$ variables are related to the symmetric functions in the $X$ variables by the transform $pk(Y(φ))= (-1)k∣φ∣-1 ∑x∈𝔽qk∣φ∣* ξ(x)pk∣φ∣d(fx) (X(fx)), (5.9)$ 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 $\frac{a}{b}\notin {ℤ}_{\ge 0}\text{.}$ Then $Λℂ=⨂f∈Φ Λℂ(X(f)).$

For $\nu \in 𝒫,$ let ${s}_{\nu }\left({Y}^{\left(\phi \right)}\right)$ be the Schur function and ${P}_{\nu }\left({X}^{\left(f\right)};t\right)$ be the Hall-Littlewood symmetric function (as in Section 2.3.3). Define $sλ=∏φ∈Θ sλ(φ) (Y(φ))and Pη=q-n(η) ∏f∈ΦPη(f) (X(f);q-d(f)), (5.10)$ 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 $\mu ,$ $n\left(\mu \right)=\sum _{i=1}^{\ell \left(\mu \right)}\left(i-1\right){\mu }_{i}\text{.}$ The ring $Λℂ=ℂ-span {sλ | λ∈𝒫Θ} -ℂ-span{Pη | η∈𝒫Φ}$ has an inner product given by $⟨sλ,sν⟩ =δλν.$

(Green [Gre1955],Macdonald [Mac1995]). The linear map $ch: R ⟶ Λℂ χλ ⟼ sλ,for λ∈𝒫Θ κη ⟼ Pη,for η∈𝒫Φ,$ 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 $𝒰=ℂ-span {κη | η(f)=∅, unless f=t-1} ⊆R$ be the subalgebra of unipotent class functions. Note that by (5.10) and Theorem 5.6, $ch(𝒰)=Λℂ (X(t-1)).$ Consider the projection $\pi :R\to 𝒰$ which is an algebra homomorphism given by $(πχλ)(g)= { χλ(g), if g∈G is unipotent, 0, otherwise, λ∈𝒫Θ.$ Then $\stackrel{\sim }{\pi }=\text{ch}\circ \pi \circ {\text{ch}}^{-1}:{\mathrm{\Lambda }}_{ℂ}\to {\mathrm{\Lambda }}_{ℂ}\left({X}^{\left(t-1\right)}\right)$ is given by $π∼(pk(Y(φ))) = π∼ ( (-1)k∣φ∣-1 ∑x∈𝔽qk∣φ∣* ξ(x)pk∣φ∣d(fx) (X(fx)) ) (by (5.9)) = (-1)k∣φ∣-1 ξ(1)pk∣φ∣1 (X(t-1))+0 = (-1)k∣φ∣-1 pk∣φ∣ (X(t-1)). (5.11)$

#### 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 ${ℋ}_{\left(n\right)}$ is commutative.

Let $U$ be the subgroup of unipotent upper-triangular matrices of $G={GL}_{n}\left({𝔽}_{q}\right)\text{.}$ Then $ch(IndUG(ψ(n))) =∑λ∈𝒫nΘht(λ)=1 sλ,where ht(λ)= max{ℓ(λ(φ)) | φ∈Θ}.$

Proof.

Let $Ψ: R ⟶ ℂ χλ ⟼ ⟨χλ,IndUG(ψ(n))⟩ andΨ∼: Λℂ⟶ch-1 R⟶Ψℂ. (5.12)$ For any finite group $H$ and $\gamma ,\chi \in R\left[H\right],$ let $1H: H ⟶ C h ⟼ 1 ,eH=1∣H∣ ∑h∈Hh,and ⟨χ,γ⟩H= 1∣H∣∑h∈H χ(h)γ(h-1).$

The proof is in six steps. (a) $\stackrel{\sim }{\mathrm{\Psi }}\left({e}_{k}\left({Y}^{\left(1\right)}\right)\right)={\delta }_{k1},$ where $1={1}_{{𝔽}_{q}^{*}},$ (b) $\mathrm{\Psi }\left({\chi }^{\lambda }\right)=\text{dim}\left({e}_{\left(n\right)}{G}^{\lambda }\right)$ for $\lambda \in {𝒫}^{\mathrm{\Theta }},$ (c) $\stackrel{\sim }{\mathrm{\Psi }}\left(fg\right)=\stackrel{\sim }{\mathrm{\Psi }}\left(f\right)\stackrel{\sim }{\mathrm{\Psi }}\left(g\right)$ for all $f,g\in {\mathrm{\Lambda }}_{ℂ}\left({Y}^{\left(1\right)}\right),$ where $1={1}_{{𝔽}_{q}^{*}}\text{.}$ (d) $\mathrm{\Psi }\circ \pi =\mathrm{\Psi },$ (e) $\stackrel{\sim }{\mathrm{\Psi }}\left(f\left({Y}^{\left(\phi \right)}\right)\right)=\stackrel{\sim }{\mathrm{\Psi }}\left(f\left({Y}^{\left(1\right)}\right)\right)$ for all $f\in {\mathrm{\Lambda }}_{ℂ}\left({Y}^{\left(\phi \right)}\right),$ (f) $\stackrel{\sim }{\mathrm{\Psi }}\left({s}_{\lambda }\right)={\delta }_{\text{ht}\left(\lambda \right)1}\text{.}$

(a) An argument similar to the argument in [Mac1995, pgs. 285-286] shows that $ch-1(ek(Y(1))) =1Gk$ (see [HRa1999, Theorem 4.9 (a)] for details). Therefore, by Frobenius reciprocity and the orthogonality of characters, $Ψ∼(ek(Y(1))) =⟨1Gk,IndUG(ψ(n))⟩= ⟨1Uk,ψ(n)⟩Uk= δk1.$

(b) Since there exists an idempotent $e$ such that ${G}^{\lambda }\cong ℂGe$ and ${\text{Ind}}_{U}^{G}\left({\psi }_{\left(n\right)}\right)\cong ℂG{e}_{\left(n\right)},$ the map $e(n)ℂGe ⟶ HomG(Gλ,ℂGe(n)) e(n)ge ⟼ γg: ℂGe ⟶ ℂGe(n) xe ⟼ xege(n)$ is a vector space isomorphism (using an argument similar to the proof of [CRe1981, (3.18)]). Thus, $Ψ(χλ) = ⟨χλ,IndUG(ψ(n))⟩ = dim(HomG(Gλ,IndUG(ψ(n)))) = dim(e(n)ℂGe) = dim(e(n)Gλ).$

(c) By (a), $\stackrel{\sim }{\mathrm{\Psi }}\left({e}_{r}\left({Y}^{\left(1\right)}\right)\right)\stackrel{\sim }{\mathrm{\Psi }}\left({e}_{s}\left({Y}^{\left(1\right)}\right)\right)={\delta }_{r1}{\delta }_{s1}\text{.}$ It therefore suffices to show that $Ψ∼(er(Y(1))es(Y(1))) =δr1δs1, (since Λℂ(Y(1))= ℂ[e1(Y(1)),e2(Y(1)),…]).$ Suppose $r+s=n$ and let $P={P}_{\left(r,s\right)}\text{.}$ Then $Ψ∼(er(Y(1))es(Y(1)))= Ψ(IndPGn(1P))= dim(e(n)ℂGeP).$ Since $T\subseteq P,$ ${e}_{P}={e}_{\left({1}^{n}\right)}{e}_{P},$ $G=\underset{v\in V}{⨆}UvU,$ and $N=WT,$ $e(n)ℂGeP=e(n) ℂGe(1n)eP=ℂ -span{e(n)we(1n)eP | w∈W}.$ If there exists $1\le i\le n$ such that $w\left(i+1\right)=w\left(i\right)+1,$ then $e(n)we(1n)= e(n)wxi,i+1 (t)e(1n)=e(n) xw(i),w(i)+1 (t)we(1n)= ψ(t)e(n)w e(1n).$ Therefore, ${e}_{\left(n\right)}w{e}_{\left({1}^{n}\right)}=0$ unless $w={w}_{\left(n\right)}\text{.}$ If $r>1$ of $s>1,$ then there exists $1\le i\le n$ such that ${x}_{i+1,i}\left(t\right)\in {P}_{\left(r,s\right)},$ so $e(n)w(n)eP= e(n)w(n)xi+1,i (t)eP=e(n) xn-i,n-i+1(t) w(n)eP=ψ(t) e(n)w(n)eP=0.$ In particular, $dim(e(n)ℂGeP)=0.$ If $r=s=1,$ then ${P}_{\left(1,1\right)}$ is upper-triangular, so $e(2)w(2)eP ≠0$ and $\text{dim}\left({e}_{\left(2\right)}ℂG{e}_{P}\right)=1,$ giving $\stackrel{\sim }{\mathrm{\Psi }}\left({e}_{r}\left({Y}^{\left(1\right)}\right){e}_{s}\left({Y}^{\left(1\right)}\right)\right)={\delta }_{r1}{\delta }_{s1}\text{.}$

(d) By Frobenius reciprocity, $⟨χλ,IndUnGn(ψ(n))⟩ = ⟨ResUnGn(χλ),ψ(n)⟩Un = ⟨ResUnGn(π(χλ)),ψ(n)⟩ = ⟨π(χλ),IndUnGn(ψ(n))⟩,$ so $\mathrm{\Psi }=\mathrm{\Psi }\circ \pi \text{.}$

(e) Induct on $n,$ using (c) and the identity $(-1)n-1pn (Y(1))=nen (Y(1))- ∑r=1n-1 (-1)r-1pr (Y(1))en-r (Y(1)), [Mac1995, I.2.11′]$ to obtain $\stackrel{\sim }{\mathrm{\Psi }}\left({p}_{n}\left({Y}^{\left(1\right)}\right)\right)=1\text{.}$ Note that $Ψ∼(pn(Y(φ))) = Ψ∼(π(pn(Y(φ)))) = Ψ∼((-1)∣φ∣n-1p∣φ∣n(X(t-1))) = Ψ∼(π(p∣φ∣k(Y(1)))) = Ψ∼(p∣φ∣k(Y(1))) = 1 = Ψ∼(pk(Y(1))).$ Since $\stackrel{\sim }{\mathrm{\Psi }}$ is multiplicative on ${\mathrm{\Lambda }}_{ℂ}\left({Y}^{\left(1\right)}\right),$ $Ψ∼(pν(Y(φ))) =1=Ψ∼(pν(Y(1))), for all partitions ν.$ In particular, since $\stackrel{\sim }{\mathrm{\Psi }}$ is linear and ${\mathrm{\Lambda }}_{ℂ}\left({Y}^{\left(\phi \right)}\right)=ℂ\text{-span}\left\{{p}_{\nu }\left({Y}^{\left(\phi \right)}\right)\right\},$ $Ψ∼(f(Y(φ)))= Ψ∼(f(Y(1))), for all f∈Λℂ(Y(φ)).$ Note that (e) also implies that $\stackrel{\sim }{\mathrm{\Psi }}$ is multiplicative on all of ${\mathrm{\Lambda }}_{ℂ}\text{.}$

(f) Note that $Ψ∼(sλ)= Ψ∼(∏φ∈Θsλ(φ)(Y(φ)))= Ψ∼(∏φ∈Θsλ(φ)(Y(1)))= ∏φ∈ΘΨ∼ (sλ(φ)(Y(1))),$ where the last two equalities follow from (e) and (c), respectively. By definition ${s}_{\nu }\left({Y}^{\left(1\right)}\right)=\text{det}\left({e}_{{\nu }_{i}^{\prime }-i+j}\left({Y}^{\left(1\right)}\right)\right),$ so $Ψ∼(sν(Y(1)))= { 1, if ℓ(ν)=1, 0, otherwise,$ implies $ch(IndUG(ψ(n))) =∑λ∈𝒫nΘ Ψ∼(sλ)sλ =∑λ∈PSnΘht(λ)=1 sλ.$

$\square$

#### 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 $\phi \in \mathrm{\Theta },$ $sh(P(φ))= λ(φ)and wt(P(φ))= ν(φ).$ 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 $IndUG(ψμ)= ⨁λ∈ℋˆμ Card(ℋˆμλ) Gλ.$

Proof.

Note that $IndUPμ(ψμ) ≅ℂPμeμ=ℂ Pμe[μ] e[μ]′,$ where $eμ=1∣U∩Lμ∣ ∑u∈U∩Lμψμ (u-1)uand e[μ]′=1∣Uμ∣ ∑u∈Uμu. (5.13)$ Thus $IndUPμ(ψμ) ≅ InfLμPμ (IndU∩LμLμ(ψμ)) ≅ InfLμPμ ( IndUμ1Gμ1 (ψ(μ1))⊗ IndUμ2Gμ2 (ψ(μ2))⊗⋯⊗ IndUμℓGμℓ (ψ(μℓ)) )$ In particular, by the definition of multiplication in $R$ (5.8), $Γμ=ch (IndUG(ψμ)) =Γμ1Γμ2 ⋯Γμℓ,where Γμi= ∑λ∈𝒫μiΘ,ht(λ)=1 sλ.$ Pieri’s rule (2.10) implies that for $\lambda \in {𝒫}_{r}^{\mathrm{\Theta }},$ $\nu \in {𝒫}_{s}^{\mathrm{\Theta }}$ and $\text{ht}\left(\nu \right)=1,$ $sλsν= ∑γ∈𝒫r+sΘ,∣ℋˆνγ/λ∣≠0 sγ,soΓμ= ∑λ∈𝒫Θ Kλμsλ,$ where $Kλμ = Card { ∅=γ0⊆ γ1⊆γ2⊆ ⋯⊆γℓ= λ | ∣ℋˆ(μi+1)γi+1/γi∣ =1 } = Card{column strict tableaux of shape λ and weight μ} = ∣ℋˆμλ∣.$ By Green’s Theorem (Theorem 5.6), ch is an isomorphism, so $IndUG(ψμ)= ch-1(Γμ)= ∑λ∈ℋˆμ ∣ℋˆμλ∣ ch-1(sλ)= ⨁λ∈ℋˆμ Card(ℋˆμλ) Gλ.$

$\square$

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

Let $\mu =\left({\mu }_{1},{\mu }_{2},\dots ,{\mu }_{\ell }\right)\models n$ and let ${P}_{\mu },$ ${L}_{\mu }$ and ${U}_{\mu }$ be as in (5.6) and (5.7). Recall that $eμ=1∣U∣ ∑u∈Uψμ (u-1)u.$

For $a\in {M}_{\mu },$ let ${T}_{a}={e}_{\mu }{v}_{a}{e}_{\mu }$ with ${v}_{a}$ as in (4.14). Then the map $ℋ(μ1)⊗ ℋ(μ2)⊗⋯⊗ ℋ(μℓ) ⟶ ℋμ T(f1)⊗ T(f2)⊗⋯⊗ T(fℓ) ⟼ T(f1)⊕(f2)⊕⋯⊕(fℓ), for (fi)∈M(μi)$ is an injective algebra homomorphism with image ${ℒ}_{\mu }={e}_{\mu }{P}_{\mu }{e}_{\mu }={e}_{\mu }{L}_{\mu }{e}_{\mu }\text{.}$

Proof.

Note that $T(f1)⊗⋯⊗ T(fℓ)= 1∣U∩Lμ∣2 ∑xi,yi∈Uμi (∏i=1ℓψμ(xi-1yi-1)) x1v(f1)y1⊗ ⋯⊗xℓv(fℓ)yℓ.$ Since $U=\left({L}_{\mu }\cap U\right)\left({U}_{\mu }\right),$ ${L}_{\mu }\cap U\cong {U}_{{\mu }_{1}}×{U}_{{\mu }_{2}}×\cdots ×{U}_{{\mu }_{\ell }},$ and ${\psi }_{\mu }$ is trivial on ${U}_{\mu },$ $T(f1)⊕(f2)⊕⋯⊕(fℓ) = 1∣U∣2 ∑x,y∈U ψμ(x-1y-1) x(v(f1)⊕v(f2)⊕⋯⊕v(fℓ))y = 1∣U∩Lμ∣2 ∑xi,yi∈Uμi ψμ(x1-1y1-1⊕⋯⊕xℓ-1yℓ-1) e[μ]′x1v(f1) y1⊕⋯⊕xℓ v(fℓ)yℓ e[μ]′,$ where ${e}_{\left[\mu \right]}^{\prime }$ is as in (5.13). Since ${L}_{\mu }\subseteq {N}_{G}\left({U}_{\mu }\right),$ the idempotent ${e}_{\left[\mu \right]}^{\prime }$ commutes with ${x}_{1}{v}_{\left({f}_{1}\right)}{y}_{1}\oplus \cdots \oplus {x}_{\ell }{v}_{\left({f}_{\ell }\right)}{y}_{\ell }$ and $T(f1)⊕(f2)⊕⋯⊕(fℓ) =e[μ]′∣L∩U∣2 ∑xi,yi∈Uμi (∏i=1ℓψμ(xi-1yi-1)) x1v(f1)y1⊕⋯⊕ xℓv(fℓ)yℓ.$ Consequently, the map multiplies by ${e}_{\left[\mu \right]}^{\prime }$ and changes $\otimes$ to $\oplus ,$ so it is an algebra homomorphism. Since the map sends basis elements to basis elements, it is also injective.

$\square$

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 ${ℋ}_{\left({\mu }_{i}\right)}$ 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 $ℋˆ(μi)= {Θ-partitions λ | ∣λ∣=μi,ht(λ)=1}$ indexes the irreducible ${ℋ}_{\left({\mu }_{i}\right)}\text{-modules.}$ Therefore, the set $ℒˆμ= ℋˆ(μ1)× ℋˆ(μ2)×⋯× ℋˆ(μℓ)= {γ=(γ1,γ2,…,γℓ) | γi∈ℋˆ(μi)} (5.14)$ indexes the irreducible ${ℒ}_{\mu }\text{-modules.}$ Identify $\gamma \in {\stackrel{ˆ}{ℒ}}_{\mu }$ with the map $\gamma :{ℒ}_{\mu }\to ℂ$ such that $yv=γ(y)v, for all y∈ℒμ, v∈ℒμγ.$

For $\gamma \in {\stackrel{ˆ}{ℒ}}_{\mu },$ the $\gamma \text{-weight}$ space ${V}_{\gamma }$ of an ${ℋ}_{\mu }\text{-module}$ $V$ is $Vγ= { v∈V | yv=γ (y)v, for all y∈ℒμ } .$ Then $V≅⨁γ∈ℒˆμ Vγ.$

Let $\lambda \in {𝒫}^{\mathrm{\Theta }}$ and $\gamma \in {\stackrel{ˆ}{ℒ}}_{\mu }\text{.}$ A column strict tableau $P$ of shape $\lambda$ and weight $\gamma$ is column strict filling of $\lambda$ such that for each $\phi \in \mathrm{\Theta },$ $sh(P(φ))= λ(φ)and wt(P(φ))= ( ∣γ1(φ)∣, ∣γ2(φ)∣,…, ∣γℓ(φ)∣ ) ,$ 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 $ℋˆγλ= { P∈ℋμλ | sh (P)=λ, wt(P)=γ } .$

For example, suppose $λ= ( (φ1) , (φ2) , (φ3) )$ and $γ= ( (φ1) , (φ2) , (φ3) ) ⊗ ( (φ1) , (φ3) ) ⊗ ( (φ1) , (φ2) ) ⊗ ( (φ2) ) .$ Then $ℋˆγλ= { ( 1 1 3 2 2 (φ1) , 1 3 4 (φ2) , 1 2 (φ3) ) , ( 1 1 3 2 2 (φ1) , 1 4 3 (φ2) , 1 2 (φ3) ) ( 1 1 2 2 3 (φ1) , 1 3 4 (φ2) , 1 2 (φ3) ) , ( 1 1 2 2 3 (φ1) , 1 4 3 (φ2) , 1 2 (φ3) ) } .$

Let ${ℋ}_{\mu }^{\lambda }$ be an irreducible ${ℋ}_{\mu }\text{-module}$ and $\gamma \in {\stackrel{ˆ}{ℒ}}_{\mu }\text{.}$ Then $dim(ℋμλ)γ= Card(ℋˆγλ).$

Proof.

By Theorem 2.3 and Proposition 2.2, $dim((ℋμλ)γ)= ⟨Resℒμℋμ(ℋμλ),ℒμγ⟩= ⟨ResPμG(Gλ),Pμγ⟩= ⟨Gλ,IndPμG(Pμγ)⟩,$ where ${P}_{\mu }^{\gamma }={\text{Inf}}_{{L}_{\mu }}^{{P}_{\mu }}\left({L}_{\mu }^{\gamma }\right)\text{.}$ Therefore, $dim((ℋμλ)γ)= cγλ,where sγ1sγ2⋯ sγℓ=∑λ∈𝒫Θ cγλsλ.$ Pieri’s rule (2.10) implies ${c}_{\gamma }^{\lambda }=\mid {\stackrel{ˆ}{ℋ}}_{\gamma }^{\lambda }\mid \text{.}$

$\square$

## 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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