Cohomology of classical algebraic groups from the functorial viewpoint

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.     

Construction of discrete series for classical -adic groups

The classification of irreducible square integrable representations of classical -adic groups is completed in this paper, under a natural local assumption. Further, this classification gives a parameterization of irreducible tempered representations of these groups. Therefore, it implies a classification of the non-unitary duals of these groups (modulo cuspidal data). The classification of irreducible square integrable representations is directly related to the parameterization of irreducible square integrable representations in terms of dual objects, which is predicted by Langlands program.

     

On the unitary dual of the classical Lie groups, Representations of

In this paper we prove that a unitary representation of whose infinitesimal character satisfies some regularity condition is infinitesimally isomorphic to an module. This is done using similar techniques as those used by the author in earlier work.

     

On the construction of irreducible characters in p-blocks with normal metacyclic defect groups

The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducible ordinary and modular characters in this type of blocks.As a by-product,k(B) and l(B) are obtained.     

A result on the Gelfand-Kirillov dimension of representations of classical groups

Let be the reductive dual pair . We show that if is a representation of (respectively ) obtained from duality correspondence with some representation of (respectively ), then its Gelfand-Kirillov dimension is less than or equal to
(respectively ).

     

Small quadratic elements in representations of the special linear group with large highest weights

For almost all p-restricted irreducible representations of the group A_n(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of A_n(K) with highest weight

On the unitary dual of

In this paper we begin a systematic study of the unitarity question for genuine representations of the group . The main result is that we find a class of unitary representations (mostly isolated) analogous to the special unipotent representations defined by D. Barbasch and D. Vogan. In particular the full unitary dual of should be obtainable from this set by complementary series.

     

On the number of irreducible characters in a 3-block with a minimal nonabelian defect group

Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs' approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k(B) and a set of irreducible Brauer characters of G having cardinality l(B). We calculate k(B) and l(B) provided that D is normal in G and D■x, y, z|x~(3n)=y~(3m)= z~3= [x, z] = [y, z] = 1, [x, y] = z(n m ≥ 2).     

On the behavior of elements of prime order from Singer cycles in representations of special linear groups

We consider extremal problems for continuous functions that are nonpositive on a closed interval and can be represented as series in Gegenbauer polynomials with nonnegative coefficients. These problems arise from the Delsarte method of finding an upper bound for the kissing number in a Euclidean space. We develop a general method for solving such problems. Using this method, we reproduce results of previous authors and find a solution in the following 11 new dimensions: 147, 157, 158, 159, 160, 162, 163, 164, 165, 167, and 173. The arising extremal polynomials are of a new type.     

A relation between the pluricomplex and the classical Green functions in the unit ball of

We give a sharp upper bound for the quotient of the pluricomplex and the classical Green functions in the unit ball of .

     

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119-171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173-177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91-117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17-22, 2001, TIFR, Mumbai, 2001, pp. 469-490] of D. Morris (Witte) on a completely different topic. We will answer that question too.     

On the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups

The aim of this paper is to study the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups. We carry over a qualitative version of the Erdös-Rényi law of large numbers for Brownian motion to simply connected step 2-nilpotent Lie groups. The method applied gives rise to a proof for qualitative results concerning the modulus of continuity of Brownian motion on simply connected step 3-resp. step 2-nilpotent Lie groups without using the Ventsel-Freidlin theory as in Baldi.     

On the asymptotic distribution of the spectrum of an operator over irreducible representations of its symmetry group

We show that a representation of a finite group in the eigenfunction space of an elliptic operator defined on a Riemannian manifold and commuting with the effective action of the group is asymptotically a multiple of the regular representation of the group.     

Geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters

In this paper we calculate the orbits of flag manifolds of the complex classical groups under the action of the sets of fixed points of Cartan involutions, and determine all the geometric parameters corresponding to representations of the classical groups with integral infinitesimal characters, which are used to discuss Arthur conjecture and the Langlands classification of the irreducible admissible representations of real classical groups(see[1])Supported by the National Natural Science Foundation of China.The second author is supported by the Post-Doctor's Foundation of China.