‎in [u‎. ‎dempwolff‎, ‎on extensions of elementary abelian groups of order $2^{5}$ by $gl(5,2)$‎, ‎textit{rend‎. ‎sem‎. ‎mat‎. ‎univ‎. ‎padova}‎, ‎textbf{48} (1972)‎, ‎359‎ - ‎364.] dempwolff proved the existence of a group of the‎ ‎form $2^{5}{^{cdot}}gl(5,2)$ (a non split extension of the‎ ‎elementary abelian group $2^{5}$ by the general linear group‎ ‎$gl(5,2)$)‎. ‎this group is the second l...

We give a new geometric construction of the big projective module in the principal block of the BGG category O, or rather the corresponding D-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheave...

We define the concept of an affinized projective variety and show how one can, in principle, obtain q-identities by different ways of computing the Hilbert series of such a variety. We carry out this program for projective varieties associated to quadratic monomial ideals. The resulting identities have applications in describing systems of quasi-particles containing null-states and can be inter...

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.

Any free presentation for the finite group G determines a central extension (R, F ) for G having the projective lifting property for G over any field k. The irreducible representations of F which arise as lifts of irreducible projective representations of G are investigated by considering the structure of the group algebra kF , which is greatly influenced by the fact that the set of torsion ele...

This article was originally published in Russian in ”Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods. Part 2” (A. M. Vershik, ed.), Zapiski Nauchnyh Seminarov POMI 240 (1997), 115–135 (this text in Russian is available via http://www.pdmi.ras.ru/znsl/1997/v240.html). As it was mentioned in the ”Journal-Ref” field this English translation was published in Journal ...