An analogue of the character formula for Hecke algebras

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

Last update: 23 April 2014

Notes and References

This is an html version of the paper An analogue of the character formula for Hecke algebras by I.V. Cherednik.

M.V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 94-95, April-June, 1987. Original article submitted March 19, 1986.

An analogue of the character formula for Hecke algebras

In this note the classical character formula of Frobenius [JKe1981] for the symmetric group $S$ is generalized to affine Hekke algebras. In the spirit of [BGG1975-2], resolutions realizing these formulas are immediately constructed. The construction was motivated by [Zel1987-2] and inspired by discussions with A. V. Zelevinskii, to whom the author expresses his deep gratitude. The author is thankful to I. M. Gel'fand for his attention to this work.

1. Suppose a $C\text{-algebra}$ $H_n$ is generated by elements $T_1,\dots ,T_{n-1}$ for which $[T_i,T_j]=0$ for $i\ne j\pm 1,$ $T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},$ $(T_i-q)(T_i+1)=0\text{.}$ Henceforth, $q$ is a power of a prime (as in [Zel1980, Rog1985]) or $q$ is taken in some defective neighborhood of $1$ in $C$ (as in [Che1986]). Adding pairwise commuting $x_1,\dots ,x_n,$ with relations $[x_i,T_j]=0$ for $i\ne j,$ $j+1,$ $x_i{T}_i-T_i{x}_{i+1}=(q-1)x_i=T_i{x}_i-x_{i+1}{T}_i,$ we obtain an affine Hecke algebra ${\mathscr{H}}_n\text{.}$ For an arbitrary family $u=(u_1,\dots ,u_n)$ we extend the left action of $H_n$ on itself to an action of ${\mathscr{H}}_n$ on $H_n$ putting $x_k\left(1\right)=q^{u_k}\text{.}$ The obtained ${\mathscr{H}}_n\text{-module}$ is denoted by $I_u\text{.}$ Next, $\ell \left(w\right)$ is the length of the reduced decomposition of $w\in S_n$ relative to $s_i=(i,i+1),$ $w\prime \ge w\stackrel{\text{def}}{\iff }l\left(w\prime \right)=l\left(w\prime w^{-1}\right)+l\left(w\right),$ $\ell \left(\text{id}\right)=0,$ $\ell \left(w\right)\le n(n-1)/2\text{.}$ On a function $f({\lambda }_1,\dots ,{\lambda }_n)$ the permutations $w\in S_n$ act by the formula $\left(wf\right)({\lambda }_1,\dots ,{\lambda }_n)=f\left(w^{-1}({\lambda }_1,\dots ,{\lambda }_n)\right),$ ${\lambda }_i\in C$

For $a,b\in C$ we will write $a\ge b\iff a\in b+{\mathbb{Z}}_{+},$ otherwise, $a<b\text{.}$ We associate with a sequence of pairs $\mu =\left(\{{\ell }_i\ge {\ell }_i^{\prime }\}\right),$ $1\le i\le r,$ the family $u^{\mu }=\left(u_k\right),$ $i\le k\le n\stackrel{\text{def}}{=}\sum _{i=1}^r(l_i-l_i^{\prime })$ of all numbers $u(i,j)$ of the form ${\ell }_i\ge u(i,j)={\ell }_i^{\prime }+j\ge {\ell }_i^{\prime }+1,$ enumerated by the rule $u_k=u(i_k,j_k),$ $k<m\iff i_k<i_m$ or $j_k-j_m<0=i_k-i_m\text{.}$ We denote by $w_{\mu }\in S_n$ the permutation of indices $u_k$ preserving $i_k$ and corresponding to the transformation ${\ell }_i^{\prime }+j\to {\ell }_i-j+1\text{.}$

Lemma 1 [Rog1985]. The family of functions ${\phi }_{s_i}=i+(i-q^{{\lambda }_2-{\lambda }_1}){(i-q)}^{-1}{T}_i$ is uniquely extended to a family $\{{\phi }_w({\lambda }_1,\dots ,{\lambda }_n),w\in S_n\}$ by the cocyclic relations ${\phi }_{xy}=y^{-1}{\phi }_x{\phi }_y$ for $xy\ge y,$ $x,y\in S_n\text{.}$ 2) The submodule $I_{\mu }=H_n{\phi }_{w_{\mu }}\left(u^{\mu }\right)$ is an ${\mathscr{H}}_n\text{-submodule}$ of $I_{u^{\mu }}$ and contains, for each $w\ge w_{\mu }$ the leading coefficient ${\stackrel{\sim }{\phi }}_w\left(u^{\mu }\right)$ of the decomposition, relative to $\lambda \to 0,$ of the function ${\phi }_w\left(u^{\nu }\right),$ $\nu =\left(\{{\ell }_i+i\lambda ,{\ell }_i^{\prime }+i\lambda \}\right)\text{.}$

2. Next, suppose that $l_j<l_i,$ $l_j^{\prime }<l_i^{\prime }$ for all $j<i\text{.}$ We associate to each permutation $\sigma \in S_r{\mu }^{\sigma }=\left(\{{\ell }_i,{\ell }_{\sigma \left(i\right)}^{\prime }\}\right)$ and the ${\mathscr{H}}_n\text{-module}$ $I_{\sigma }=I_{{\mu }^{\sigma }}\text{.}$ If for some $i,$ ${\ell }_i-{\ell }_{\sigma \left(i\right)}^{\prime }<0,$ then ${\mu }^{\sigma }\stackrel{\text{def}}{=}\varnothing ,$ $I_{\sigma }\stackrel{\text{def}}{=}0\text{.}$ Put $w_{\sigma }=w_{{\mu }^{\sigma }},$ $u^{\sigma }=u^{{\mu }^{\sigma }}\text{.}$ We will write $\sigma \Rightarrow \tau$ if $\ell \left(\sigma \right)=\ell \left(\tau \right)+1$ and $\tau$ is obtained by dropping some ${\sigma }_i$ from the reduced decomposition of $\sigma$ relative to ${\sigma }_i=(i,i+1)\in S_r\text{.}$

Lemma 2. 1) If $\sigma \Rightarrow \tau ,$ then there exists a unique permutation $S_n\ni w_{\sigma ,\tau }\ge w_{\tau }$ for which $w_{\sigma ,\tau }\left(u^{\tau }\right)=w_{\sigma }\left(u^{\sigma }\right)\text{.}$ 2) The condition ${\phi }_{w_{\sigma }}\left(u^{\sigma }\right)\Rightarrow {\stackrel{\sim }{\phi }}_{w_{\sigma ,\tau }}\left(u^{\tau }\right)$ uniquely determines an embedding of ${\mathscr{H}}_n\text{-modules}$ ${\rho }_{\sigma ,\tau }:I_{\sigma }\Rightarrow I_{\tau }\text{;}$ ${\rho }_{\sigma ,\tau }\left(I_{\sigma }\right)\ne I_{\tau },$ if $I_{\tau }\ne 0\text{.}$ 3) Conversely, if $\ell \left(\sigma \right)=\ell \left(\tau \right)+1$ and $I_{\tau }\ne 0,$ then there exists a nonzero ${\mathscr{H}}_n\text{-homomorphism}$ between $I_{\sigma }$ and $I_{\tau },$ $\rho \iff \sigma \Rightarrow \tau ,$ $\rho =c{\rho }_{\sigma ,\tau },$ $c\in {ℂ}^{*}\text{.}$

A family $\sigma \Rightarrow \sigma \prime \Rightarrow \tau ,$ $\sigma \Rightarrow \tau \prime \Rightarrow \tau ,$ $\sigma \prime \ne \tau \prime$ will be called a square. Each triple $\sigma \prime \Rightarrow \tau \Leftarrow \tau \prime ,$ $\sigma \prime \ne \tau \prime$ can be extended to a square.

Proposition 3. 1) For each square ${\rho }_{\sigma \prime ,\tau }{\rho }_{\sigma ,\sigma \prime }\left(I_{\sigma }\right)={\rho }_{\tau \prime ,\tau }{\rho }_{\sigma ,\tau \prime }\left(I_{\sigma }\right)\text{.}$ 2) The image ${\bar{I}}_{\sigma }$ of each $I_{\sigma }$ $(\sigma \in S_r)$ in $I_{\mu }$ does not depend on the choice of a chain $\sigma \Rightarrow \dots \Rightarrow {\sigma }_0$ and the corresponding sequence of embeddings.

Let $\sigma =\prod _{k=1}^l{\sigma }_{i_k}={\sigma }_{i_l}\cdots {\sigma }_{i_1}$ be some reduced decomposition ${\sigma }^p=\prod _{k=1}^p{\sigma }_{i_k}\in S_r\text{.}$ We associate to each ${\sigma }_{i_p}$ the element ${\stackrel{\sim }{\sigma }}_{i_p}\in S$ permuting the subfamilies $({\stackrel{\sim }{\ell }}_{i+1}^{\prime },\dots ,{\stackrel{\sim }{\ell }}_i^{\prime }+1)$ and $({\stackrel{\sim }{\ell }}_{i+1},\dots ,{\stackrel{\sim }{\ell }}_{i+1}+1)$ in $w_{\tau }\left(u^{\tau }\right)$ for $\tau ={\sigma }^{p-1},$ $\stackrel{\sim }{\mu }={\mu }^{\tau }=\left(\{{\stackrel{\sim }{\ell }}_i,{\stackrel{\sim }{\ell }}_i^{\prime }\}\right)\text{.}$ Put $\stackrel{\sim }{\sigma }=\prod _{k=1}^l{\stackrel{\sim }{\sigma }}_{ik}\text{.}$ We can verify that $\stackrel{\sim }{\sigma }$ does not depend on the choice of $\sigma$ and $\stackrel{\sim }{\sigma }\stackrel{\text{def}}{=}\stackrel{\sim }{\sigma }{w}_{\mu }\ge w_{\mu }\text{.}$

Corollary 4. If one puts ${\omega }_{\sigma }=w_{\sigma }^{-1}\stackrel{\sim }{\sigma },$ then $\stackrel{\sim }{\sigma }\ge {\omega }_{\sigma },$ ${\stackrel{\sim }{I}}_{\sigma }=H_n{\phi }_{\stackrel{\sim }{\sigma }}\left(u^{\mu }\right)$ for each $\sigma \in S_r\text{.}$ The isomorphism of $I_{\sigma }$ and ${\stackrel{\sim }{I}}_{\sigma }$ mapping ${\phi }_{w_{\sigma }}\left(u^{\sigma }\right)$ into ${\phi }_{\stackrel{\sim }{\sigma }}\left(u^{\mu }\right)$ is induced by multiplication of $H_n$ on the right by ${\phi }_{{\omega }_{\sigma }}\left(u^{\mu }\right)\text{.}$ If $I_{\sigma }\ne 0,$ then ${\stackrel{\sim }{I}}_{\sigma }\subset {\stackrel{\sim }{I}}_{\tau }\iff \sigma \Rightarrow \dots \Rightarrow \tau$ for some chain.

Proposition 5 [BGG1975-2]. On the set of pairs $\sigma \Rightarrow \tau$ there exists a function $\epsilon (\sigma ,\tau )=\pm 1$ for which $\epsilon (\sigma \prime ,\tau )\epsilon (\sigma ,\sigma \prime )=-\epsilon (\tau \prime ,\tau )\epsilon (\sigma ,\tau \prime )$ on each square.

Put $V_p=\underset{\ell \left(\sigma \right)=p}{⨁}{\stackrel{\sim }{I}}_{\sigma }\text{.}$ Let ${\nu }_{\sigma ,\tau }:V_p\to V_{p-1}$ be the homomorphism, defined for $\sigma \Rightarrow \tau ,$ $\ell \left(\sigma \right)=p,$ inducing the natural embedding ${\stackrel{\sim }{I}}_{\sigma }\subset {\stackrel{\sim }{I}}_{\tau }\subset I_r$ and mapping each ${\stackrel{\sim }{I}}_{\sigma \prime }\subset V_p$ for $\sigma \prime \ne \sigma$ into zero. Put $d_p=\sum _{\sigma \Rightarrow \tau }\epsilon (\sigma ,\tau ){\nu }_{\sigma ,\tau },$ $l\left(\sigma \right)=p\text{.}$ We denote by $d_0$ the homomorphism of $I_{\mu }=V_0$ onto its only irreducible quotient module $V_{-1}\ne 0$ (cf. [Zel1980]) generalizing the representation of $S_n$ associated with Jung's skew scheme corresponding to $\mu$ [JKe1981, Che1986, Che1986-2]. Let $V_p=0$ for $p>r(r-1)/2,$ $p<-1\text{.}$

Theorem 6. The sequence $\{V_p,d_p\}$ is exact.

3. Remarks. We will give an example of the function $\epsilon \text{.}$ A reduced decomposition of $\sigma \in S_r$ is said to be canonical if: a) for $1\le i<r,$ the decomposition remains reduced after eliminating all ${\sigma }_1,\dots ,{\sigma }_i\text{;}$ b) for adjacent ${\sigma }_i{\sigma }_j$ in the decomposition we always have $i>j$ when $j\ne i\pm 1\text{.}$ The author's attention was attracted to such decompositions by A. N. Kirillov. Put $\epsilon (\sigma ,\tau )={(-1)}^{k+\pi },$ where ${\sigma }_{i_k}$ is the transposition eliminated from the canonical decomposition of $\sigma$ in the passage to $\tau$ $\text{(}k$ is its number in the decomposition), $\pi$ is the number of permutations of adjacent pairs ${\sigma }_i{\sigma }_j\to {\sigma }_j{\sigma }_i$ for $i\ne j\pm 1$ (the replacements ${\sigma }_i{\sigma }_{i\pm 1}{\sigma }_i\to {\sigma }_{i\pm 1}{\sigma }_i{\sigma }_{i\pm 1}$ are not counted) used to transform the obtained reduced decomposition for $\tau$ to the canonical one.

2)To prove the theorem, using the results of [Che1986, Che1986-2] on branching of special bases in $V_{-1},$ we impose an induction restriction on ${\mathscr{H}}_{n-1}\subset {\mathscr{H}}_n\text{.}$ Here, $\{V,d\}$ splits into a direct sum of some sequences $\{V^i,d^i\}$ for ${\mu }^i$ in which ${\ell }_i$ are replaced by ${\ell }_i-1\text{.}$ For an admissible ${\mu }^i$ (satisfying the same inequalities as $\mu \text{),}$ after eliminating all ${\stackrel{\sim }{I}}_{\sigma }$ whose construction is not compatible with the passage ${\ell }_i\to {\ell }_i-1,$ the sequence $\{V^i,d^i\}$ coincides with the sequence of the theorem for ${\mathscr{H}}_{n-1}$ instead of ${\mathscr{H}}_n$ and ${\mu }^i$ instead of $\mu \text{.}$ If ${\mu }^i$ is not admissible, then it turns out that the sequence $\{V^i,d^i,V_{-1}^i\stackrel{\text{def}}{=}0\}$ is exact. In the proof of this fact one verifies that for each $\sigma$ one of the embeddings of the form ${\nu }_{\sigma ,\tau }^i$ or ${\nu }_{\tau ,\sigma }^i$ is an isomorphism for a suitable $\tau \text{.}$

3) All constructions of this note, as well as the theorem, are extended verbatim to the degenerate algebra ${\mathscr{H}}_n$ $(q\to 1)$ (cf. [Che1986, Che1986-2, Dri1986-2]). For this algebra, $T_i$ can be identified with $s_i$ and $x_i{s}_i-s_i{x}_{i+1}=1=s_i{x}_i-x_{i+1}{s}_i\text{.}$ Respectively, one has to put $x_k\left(1\right)=u_k,$ ${\phi }_{s_i}=1+({\lambda }_2-{\lambda }_1)s_i\text{.}$ Then $I_{\mu }$ is isomorphic as an $S_n\text{-module}$ to the representation induced from the identity representation of the subgroup $\prod _{i=1}^r{S}_{l_i-l_i^{\prime }}\subset S_n\text{;}$ $V_{-1}$ for integers $\left\{{\ell }_i\right\}$ $V_{-1}$ corresponds to Jung's skew scheme [JKe1981] represented by cells with the set of centers $(i,j)\subset Z^2,$ $0\le i\le r-1,$ ${\ell }_{r-i}^{\prime }+i<j\le {\ell }_{r-i}+i\text{.}$ Thus, the theorem, indeed, generalizes the character formula of Frobenius. A replacement of $\left(u_k\right)$ by $(-u_k)$ in all constructions results in a similar "antisymmetric" resolution for the ${\mathscr{H}}_n\text{-module}$ corresponding to the scheme obtained from the scheme $\mu$ by the reflection in the diagonal $i=j\text{.}$ This is also true for $q\ne 1\text{.}$ The statement and the proof of Theorem 6 are extended to representations of quantum $R\text{-algebras}$ of the A series corresponding to $\mu$ [Che1986-2, Dri1986-2] (cf. [Che1986-3]).

Literature cited

[JKe1981] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[BGG1975-2] I.N. Bernstein, I.M. Gel'fand, and S.I. Gel'fand, Publ. of 1971 Summer School in Math., Budapest (1975), pp. 21-64.

[Zel1987-2] A.V. Zelevinskii, Funktsion. Anal. Prilozhen., 21, No. 2, 74-75 (1987).

[Zel1980] A. Zelevinsky, Induced representations of $𝔭\text{-adic}$ groups II: On irreducible representations of $GL\left(n\right)$, Ann. Sci. École Norm. Sup. Ser. (4) 13 (1980) 165–210.

[Rog1985] J. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985) 443–465. MR 86j:22028

[Che1986] I.V. Cherednik, in: Group-Theoretic Methods in Physics. Proceedings of the 3rd Int. Sem. [in Russian], Nauka, Moscow (1986).

[Che1986-2] I.V. Cherednik, Funktsion. Anal. Prilozhen., 20, No. 1, 87-88 (1986).

[Dri1986-2] V.G. Drinfel'd, Funktsion. Anal. Prilozhen., 20, No. 1, 69-70 (1986).

[Che1986-3] I.V. Cherednik, Dokl. Akad. Nauk SSSR, 291, No. 1, 49-53 (1986).

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