George Lusztig

Canonical bases arising from quantized enveloping algebras

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see ?2) bases of PBW type, since for v = 1, they specialize to bases of U+ of the type provided by the Poincare see however ? 12.)

Representations of reductive groups over finite fields

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Quantum groups at roots of 1

We extend to the not necessarily simply laced case the study [8] of quantum groups whose parameter is a root of 1.

Quivers, perverse sheaves, and quantized enveloping algebras

1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of multiplication with restriction 9. Rank 2 10. Definition of the canonical basis B of U 11. Properties of the canonical basis B of U 12. The variety AV 13. Singular supports 14. Example: graphs of type A, D, E 15. Example: graphs of affine type A 16. Graphs with a cyclic group action

Cuspidal local systems and graded Hecke algebras, I

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On the Modular Representations of the General Linear and Symmetric Groups

Let V be an n-dimensional vector space over C with basis X 1, X 2,..., X n and let T r = V ⊗ V ⊗ ... ⊗ V (r factors). T r is a module both for GL n (C), via its action on V, and for the symmetric group S r by place permutations.

Cells in affine Weyl groups, II

This paper is a continuation of [S]. The main theme of [S] was the study of a numerical function w --f a(w) on a Coxeter group W which in the case of Weyl groups is closely related to the Gelfand-Kirillov dimension of certain modules over the corresponding enveloping algebra. In [S], this function was defined purely in terms of multiplication in the Hecke algebra. Several properties of this function were proved in [S] only for Weyl groups. We shall prove them here for a larger class of Coxeter groups including the afline Weyl groups. We show that if W is an affine Weyl group, then W contains only finitely many left cells; we also show that each two sided cell of W which is finite, carries a “square integrable” representation of the corresponding Hecke algebra. One of the main themes of this paper is provided by certain distinguished involutions of W, one in each left cell. In [ 11, Joseph shows that for each left cell of a Weyl group, the function y H Z(y) 26(y) (I= length, 6 = degree of the polynomial P,, of [2]) reaches its minimum at a unique element of that left cell, the “Duflo involution.” We show that this minimum value is a(y) for any y in the left cell; from this, Joseph’s conjecture V4 in [ 1 ] follows easily. We also prove the analogous result for ahme Weyl groups. The proof is an adaptation of the proof of the inequality a(w) < Z(w) which was shown to me by T. A. Springer (see 1.2). An important role in our proofs is played by a ring J which has a basis 8, (w E W) over Z and in which the structure constants are certain 20 integers yX, y,z = ( l)a(z)~, ,-“.= (x, y, z E W) where c,,~,~ are defined in [ 5, 5.11 in terms of multiplication of the elements C, of [2] in the Hecke algebra. This may be regarded as an asymptotic version of the Hecke algebra H, one of our main results is a comparison theorem (2.8) between H and J. I want to thank A. Joseph, T. A. Springer, and D. Vogan for some very useful discussions. The topics in this paper were the subject of lccturcs given at the University of Rome II and the University of Sydney. 536 0021-8693/87

Total Positivity in Reductive Groups

3.00

Hecke algebras and Jantzen's generic decomposition patterns

An invertible n×n matrix with real entries is said to be totally ≥0 (resp. totally >0) if all its minors are ≥0 (resp. >0). This definition appears in Schoenberg’s 1930 paper [S] and in the 1935 note [GK] of Gantmacher and Krein. (For a recent survey of totally positive matrices, see [A].)

Fixed point varieties on affine flag manifolds

The purpose of this paper is to describe certain patterns arising in the study of the Hecke algebra associated to an tine Weyl group; these patterns are (conjecturally) q-analogues of the generic decomposition patterns (due to Jantzen [5]) which involve multiplicities in the “Weyl modules” of a simply connected almost simple algebraic group G over an algebraically closed field k of characteristic p > 1. We shall recall some of Jantzen’s results. We fix a maximal torus T and a Bore1 subgroup B of G containing T. Let S be the lattice of characters of T, and let V= 27 @ R. An element of V is said to p-regular if it does not lie on any of the hyperplanes H&,, = {AE VlG(A+p)=np}, where a’: V+IR is a coroot, n is an integer and p E K is half the sum of positive roots (with respect to B).‘ The p-regular elements form an open set whose connected components (alcoves) are open simplices. An element d E X is said to be dominant if a’@) > 0 for any simple coroot a’i (i = l,..., I). For such A there is an irreducible rational G-module L, (unique up to isomorphism) with highest weight 1. One can describe L, as a quotient of the Weyl module V.\ with highest weight ,I. We shall assume that p is sufficiently large; in particular, each alcove contains elements of 37. An alcove is said to be dominant if it containts some dominant element of 227. The set of dominant alcoves has a standard partial order < (see [4, lo]). Let us associate to each alcove A a weight 1, E 27 n A in such a way that, whenever an alcove B is the mirror image of an alcove A in a hyperplane H&+,, the weight & is the mirror image of 1, in H,,“. Consider the simplex d c V bounded by the hypevlanes Ha,,O,..., H,,,, HGc,,, (where ii,, is the highest coroot). A contains p’ alcoves. Let B <A be two alcoves. We translate simultaneously B, A to two new alcoves B’ <A’ in such a way that A’ is inside A, far from its walls.