A q-analogue of U(g[(N+1)), Hecke algebra, and the Yang-Baxter equation
We study for g=g[(N+1) the structure and representations of the algebra Ŭ(g), a q-analogue of the universal enveloping algebra U(g). Applying the result, we construct trigonometric solutions of the Yang-Baxter equation associated with higher representations of g.
Ring-Theoretic Properties of Certain Hecke Algebras
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method going back to Mazur [M] allows one to show that these algebras are Gorenstein, but for the complete intersection property a new approach is required. The methods of this paper are related to those of Chapter 3 of [W2]. The methods of Section 3 of this paper are based on a previous approach of one of us (A.W.). We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument as well as of the argument of Chapter 3 of [W2] and we would like to thank him for allowing us to reproduce these in the appendix to this paper. R. T. would like to thank A. W. for his invitation to collaborate and for sharing his many insights into the questions considered. R. T. would also like to thank Princeton University, Universite de Paris 7 and Harvard University for their hospitality during this collaboration. A. W. was supported by an NSF grant.
Affine Hecke algebras and their graded version
0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the "special case" where c(s) is independent of s and the coroots generate a direct summand. In this "special case," the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.
Hecke algebras at roots of unity and crystal bases of quantum affine algebras
AbstractWe present a fast algorithm for computing the global crystal basis of the basic $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$ -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at anth root of 1.
D-Modules, Perverse Sheaves, and Representation Theory
D-Modules and Perverse Sheaves.- Preliminary Notions.- Coherent D-Modules.- Holonomic D-Modules.- Analytic D-Modules and the de Rham Functor.- Theory of Meromorphic Connections.- Regular Holonomic D-Modules.- Riemann-Hilbert Correspondence.- Perverse Sheaves.- Representation Theory.- Algebraic Groups and Lie Algebras.- Conjugacy Classes of Semisimple Lie Algebras.- Representations of Lie Algebras and D-Modules.- Character Formula of HighestWeight Modules.- Hecke Algebras and Hodge Modules.
Hecke algebras and Schur algebras of the symmetric group
The Iwahori-Hecke algebra of the symmetric group Cellular algebras The modular representation theory of $\mathcal {H}$ The $q$-Schur algebra The Jantzen sum formula and the blocks of $\mathcal H$ Branching rules, canonical bases and decomposition matrices Appendix A. Finite dimensional algebras over a field Appendix B. Decomposition matrices Appendix C. Elementary divisors of integral Specht modules Index of notation References Index.
Some examples of square integrable representations of semisimple p-adic groups
We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple p-adic group.
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