Let K be a field of characteristic p> 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first « — 1 letters, and D k the irreducible ATn-module corresponding to a (/^-regular) partition X of n. In [9] we described the socle of the restriction D x [T and obtained a number of other results

Introduction and statement of results Finite primitive permutation groups of rank 3 have been the subject of much study in the past twenty years, leading on the one hand to interesting new groups (for instance, some sporadic groups) and on the other to new techniques in the theory of permutation groups. It is readily seen that if G is a primitive rank 3

Abstract. Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials (q-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from “scratch”, in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Schützenberger’s positive formula for the Kostka-Foulkes poynomials in the type A case. The classical theory of Hall-Littlewood polynomials and the Kostka-Foulkes polynomials appears in the monograph of I.G. Macdonald [Mac]. The Hall-Littlewood polynomials form a basis of the ring of symmetric functions and the Kostka-Foulkes polynomials are the entries of the transition matrix between the Hall-Littlewood polynomials and the Schur functions.

Abstract. In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such a formula was obtained recently in [8]. We also use our techniques to study the structure of the space of smooth vectors in the minimal representation. In the second part we consider groups G as above defined over a global field K. In this situation we describe the form of the automorphic functional on the minimal representation of the corresponding adelic group.

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this paper describes the multiplicities of some composition factors in

Let G be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroup P of G acts on its unipotent radical Pu, or on pu, the Lie algebra of Pu, with only a finite number of orbits. The proof proceeds in two parts. First we obtain a reduction to the case of general linear groups. In a second step, a solution for these is achieved by studying the representation theory of a particular quiver with certain relations. Furthermore, for the general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.

Abstract. We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp 4 to GL4 follows from our result as a special case. 1.

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We prove that for a simple simply connected quasi-split group of type 2, 3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F 2 (resp. vcd F 2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C 2 )-field, in particular C(x, y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type D 4 , E 6 , E 7 .