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Representations of Coxeter groups and Hecke algebras. (English) Zbl 0499.20035
In this paper of exceptional importance, the authors have been able to “explain” many combinatorial phenomena that are associated to a Weyl group in different contexts. The key discovery is that of a set of certain polynomials \(P_{x,y}\) in one variable with integral coefficients associated to a pair \((x,y)\) of elements of a Coxeter group \(W\). These polynomials are used extensively to (1) construct certain representations of the Hecke algebra of \(W\) thereby obtaining important information on representations of \(W\), (2) give a formula (conjecturally) for the multiplicities in the Jordan-Hölder series of Verma modules or equivalently for the formal characters of irreducible highest weight modules (extending the celebrated Weyl character formula to “nondominant” weights), (3) give a complete description for the inclusion-relations between various primitive ideals in the enveloping algebra of a complex semisimple Lie algebra \(\mathfrak g\); in case \(\mathfrak g\) is of type \(A_n\), this is further tied up with the dimensions of certain representations of the corresponding Weyl group (via the “Jantzen conjecture” – cf. a paper by A. Joseph [Lect. Notes Math. 728, 116–135 (1979; Zbl 0422.17004)], (4) give a measure of the failure of local Poincaré duality in the geometry of Schubert cells in flag varieties. (In a later paper [D. Kazhdan and G. Lusztig, Proc. Symp. Pure Math. 36, 185–203 (1980; Zbl 0461.14015)] the authors give a more precise interpretation of the coefficients of \(P_{x,y}\) in terms of a certain cohomology theory called “middle intersection cohomology” associated with the geometry of Schubert cells.)
Considering the various topics involved and their importance, it seems worthwhile to give a detailed review in order to give some idea of the wealth of information contained in this paper.
We first describe the combinatorial setup involved. Let \((W,S)\) be a Coxeter group. Let \(\mathbb Z[q^{1/2},q^{-1/2}]\) be the ring of Laurent polynomials in the indeterminate \(q^{1/2}\) over \(\mathbb Z\). Let \(\mathcal H\) be the free \(\mathbb Z[q^{1/2},q^{-1/2}]\)-module with \(\{T_y\mid y\in W\}\) as a basis; the multiplication in \(\mathcal H\) is given by: for \(s\in S\), \(y\in W\), \(T_s\cdot T_y=T_{sy}\) if \(l(sy)\geq l(y)\) and \(T_s\cdot T_y=(q-1)T_y+q\cdot T_{sy}\) if \(l(sy)\leq l(y)\). (Classically, one considers \(\mathbb Z[q]\)-coefficients only; the algebra \(\mathcal H\) thus obtained, called the Hecke algebra of \(W\), is isomorphic to the space of intertwining operators on the “\(1_B^G\)”-representation of a finite Chevalley group with \(W\) as the Weyl group.)
It can be seen that \(T_y\) is invertible in \(\mathcal H\) and so \(\mathcal H\) has an involution \({}^-\) under which \(T_y\) goes to \(T_{y^{-1}}^{-1}\) and \(q^{1/2}\) goes to \(q^{-1/2}\). The main discovery of the paper can now be stated as the theorem: For any \(y\in W\), there is a unique element \(C_y\in\mathcal H\) such that (i) \(\overline C_y=C_y\) and (ii) \(C_y=\sum_{x\in W}(-1)^{l(x)+l(y)}\cdot(q^{1/2})^{l(y)}\cdot q^{-l(x)}\overline P_{x,y}\cdot T_x\), where \(P_{x,y}\in\mathbb Z[q]\) with \(P_{y,y}=1\) and \(\deg P_{x,y}\leq(l(y)-l(x)-1)/2\) if \(x\mathop{<}\limits_{\neq}y\) (\(\leq\) is the Bruhat ordering on \(W\)) and \(P_{x,y}=0\) otherwise.
The authors give an inductive formula for \(P_{x,y}\); however, no closed formula is available as yet. It is conjectured that the coefficients of \(P_{x,y}\) are nonnegative; the authors have proved it in the case of Weyl groups and affine Weyl groups by showing them to be dimensions of certain cohomology groups [cf. the authors, op. cit.].
We now describe the various applications of the polynomials \(P_{x,y}\).
(1) Representations of \(\mathcal H\): In order to obtain certain representations of \(H\), the authors introduce the notion of a \(W\)-graph as follows: It is a graph \(\Gamma\) without loops such that to each vertex \(x\in\Gamma\) is associated a subset \(I_x\) of \(S\) (the set of simple reflections in \(W\)) and to each edge \((x,y)\) is associated a nonzero integer \(\mu(x,y)\) which is required to satisfy certain compatibility conditions. (These conditions ensure that one can define a representation of \(\mathcal H\).) Define a preorder \(x\leq_\Gamma y\) on vertices of \(\Gamma\) by: \(x\leq_\Gamma y\) if there exist \(x=x_0,x_1,\dots,x_n=y\in\Gamma\) such that for all \(i\), \((x_i,x_{i+1})\) is an edge of \(\Gamma\) with \(I_{x_i}\not\subset I_{x_{i+1}}\). Let \(\sim\) be the equivalence relation associated with \(\leq_\Gamma\) (i.e. \(x\sim y\) if \(x\leq_\Gamma y\) and \(y\leq_\Gamma x\)). Then each equivalence class considered as a full subgraph of \(\Gamma\) and the assignments “\(I_x\) and \(\mu(x,y)\)” coming from \(\Gamma\) is a \(W\)-graph itself and thus one gets many representations of \(\mathcal H\) (e.g., the “reflection” representation of \(\mathcal H\) can be obtained in this way). The authors construct a \(W\)-graph from the polynomials \(P_{x,y}\) in the following way: The set \(W\) is the set of vertices and \((x,y)\) is an edge if either \(x\mathop{<}\limits_{\neq}y\) with \(\deg P_{x,y}=(l(y)-l(x)-1)/2\) or \(y<x\) with \(\deg P_{y,x}=(l(x)-l(y)-1)/2\) (one denotes such pairs by \(x\prec y\) or \(y\prec x\) as the case may be). For \(x\in W\), assign \(I_x=L_x=\{s\in S\mid l(sx)\leq l(x)\}\) and for an edge \((x,y)\), assign \(\mu(x,y)=\) leading coefficient of \(P_{x,y}\) (or \(P_{y,x}\) as the case may be). The authors show that this gives a \(W\)-graph and the corresponding representation is in fact the left-regular representation of \(\mathcal H\). The equivalence classes of this \(W\)-graph are called left cells (“left” because the set \(L_x\) involves multiplication on the left by the elements of \(S\)). One has a similar \(W\)-graph by considering multiplication on the right and a \(W\times W^0\)-graph (\(W^0\) is the opposite group) by considering the left and right multiplications simultaneously. The equivalence classes are called right cells and two-sided cells, respectively. It turns out that the configuration of these cells forms important combinatorial data from which much information can be obtained. In case \(W=S_n\), the representations of \(W\) obtained from left cells of \(W\) by specializing \(q=1\) cover all complex representations equipped with a distinguished basis.
(2) Characters of highest weight representations of a complex semisimple Lie algebra \(\mathfrak g\): Fix a Cartan subalgebra \(\mathfrak h\) and a set of simple roots \(\Pi\) for the root system of \((\mathfrak g,\mathfrak h)\). Let \((W,S)\) be the corresponding Coxeter system. For \(x\in W\), let \(M_x\) be the Verma module with highest weight \(x\rho-\rho\) (\(\rho\) is the half-sum of positive roots) and let \(L_x\) denote the (unique) irreducible quotient of \(M_x\). For \(x,y\in W\), let \(\text{mtp}(x,y)\) be the multiplicity with which \(L_y\) occurs in a Jordan-Hölder series of \(M_x\). It is then known that \(\text{mtp}(x,y)\neq 0\) if and only if \(x\leq y\) (\(\leq\) being the Bruhat ordering). The problem of determining \(\text{mtp}(x,y)\) has been considered by several people (e.g., [J. Lepowsky and the reviewer, J. Algebra 49, 512–524 (1977; Zbl 0381.17004); J. C. Jantzen, Moduln mit einem höchsten Gewicht. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0426.17001)]). The authors conjecture that \(\text{mtp}(x,y)=P_{x,y}(1)\). (This conjecture has been proved recently by Brylinski and Kashiwara and, independently, by Beilinson and Bernstein.) This has an equivalent formulation in terms of the formal character of \(L_x\). (Recall: If \(x=\text{id}\) then one has the Weyl character formula for a finite-dimensional representation of \(\mathfrak g\).) In his talk at the AMS Santa Cruz Conference on Finite Groups [G. Lusztig, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 313–317 (1980; Zbl 0453.20005)] the second author proposed a modular analogue analogue the above conjecture from which the character formula of a rational irreducible representation of a Chevalley group \(G\) over an algebraically closed field of positive characteristic can be obtained.
(3) Theory of primitive ideals in the enveloping algebra \(U(\mathfrak g)\) of a complex semisimple Lie algebra \(\mathfrak g\): A two-sided ideal \(I\) in \(U(\mathfrak g)\) is said to be “primitive” if it is the annihilator of an irreducible \(\mathfrak g\)-module. One is then interested in the space \(X\) of primitive ideals (equipped with the Jacobson topology). As every irreducible \(\mathfrak g\)-module has a central character, one gets a map \(X\rightarrow Z(\hat{\mathfrak g})\) (\(=\text{Hom}_{\mathbb C-\text{alg}}(Z(\mathfrak g),{\mathbb C})\) where \(Z(\mathfrak g)\) is the centre of \(U(\mathfrak g)\)). By a well-known theorem of Harish-Chandra \(Z(\hat{\mathfrak g})\simeq\mathfrak h^\ast|_W\). Let \(\Lambda\) be an equivalence class. One is interested in the fibre \(X_\Lambda\) over \(\Lambda\). Let \(\lambda\in\Lambda\) and \(J(\lambda)=\text{Ann}\,L(\lambda)\), where \(L(\lambda)\) is the irreducible \(\mathfrak g\)-module with highest weight \(\lambda\). Then \(J(\lambda)\in X_\lambda\) and in fact a deep result of Duflo asserts that every element of \(X_\Lambda\) is of this form. There are various results known about the structure of fibres, “similarity” of two fibres, etc. The most important problem is to determine the inclusion relation between two elements of the same fibre. Known results by Borho, Duflo, Jantzen, Joseph, Vogan, etc. give sufficient conditions for it. However, since the conjecture “\(\text{mtp}(x,y)=P_{x,y}(1)\)” is proved to be true, the left cells of \(W\) determine the inclusion completely. To be more precise, let \(\lambda\) be an antidominant integral element. Let \(\Lambda\) be the equivalence class to which it belongs. Then every element of \(X_\Lambda\) is of the form \(J(x\cdot\lambda)\) where \(x\cdot\lambda=x(\lambda+\rho)-\rho\). Then \(J(x\cdot\lambda)\subseteq J(y\cdot\lambda)\) if and only if \(y\leq_Lx\). (The inclusion relations in other fibres can be written down with the help of a “suitable” subgroup \(W\) which corresponds to a sub-root-system.) It may be recalled that the primitive spectrum \(X\) is related to nilpotent orbits in \(\mathfrak g\).
(4) Geometry of Schubert cells in flag varieties: Let \(G\) be a complex semisimple algebraic group, \(B\) a Borel subgroup and \(G/B\) be the corresponding flag variety. \(B\) acts on \(G/B\) and one has the Bruhat decomposition \(G/B=\bigcup_{x\in W}BxB\). For \(y\in W\), let \(X(y)=\overline{ByB}=\bigcup_{x\leq y}BxB\) (\(\leq\) is the Bruhat ordering). Let \(e_x\) be the point \(xB\in G/B\). Then one is interested in determining the nature of the singularity at \(e_x\) when considered as a point of \(X(y)\;(x\leq y)\). The authors show that the condition “\(P_{x',y}\equiv 1\) for all \(x\leq x'\leq y\)” is related to the singularity of \(e_x\) in \(X(y)\). A closer tie-up is given in a later paper [the authors, loc. cit.].
Considering the central role played by the polynomials \(P_{x,y}\) in above-mentioned contexts, it is desirable to have an explicit knowledge of their coefficients in terms of certain combinatorial data. The relation \(x\prec y\) is another key notion which should be investigated further.

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G05 Representation theory for linear algebraic groups
20C08 Hecke algebras and their representations
17B35 Universal enveloping (super)algebras
20G15 Linear algebraic groups over arbitrary fields
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
14L30 Group actions on varieties or schemes (quotients)
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References:
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