Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typo

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S 15G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient // for arbitrary simple G and give a necessary and sufficient condition for //G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G2 is worked out in detail

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S/G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient \overline{S}//G for arbitrary simple G and give a necessary and sufficient condition for S//G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G_2 is worked out in detail

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

The orbit structure of Dynkin curves

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \cB be the variety of Borel subgroups of G and let e in Lie G be nilpotent. There is a natural action of the centralizer C_G(e) of e in G on the Springer fibre \cB_e = {B' in \cB | e in Lie B'} associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \cB_e is a Dynkin curve. We give a complete description of the C_G(e)-orbits in \cB_e. In particular, we classify the irreducible components of \cB_e on which C_G(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.Comment: 12 pages, to appear in Math

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

Catenarity in quantum nilpotent algebras

In this paper, it is established that quantum nilpotent algebras (also known as CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime ideals between any two given prime ideals $P \subsetneq Q$ have the same length. This is achieved by proving that the prime spectra of these algebras have normal separation, and then establishing the mild homological conditions necessary to apply a result of Lenagan and the first author. The work also recovers the Tauvel height formula for quantum nilpotent algebras, a result that was first obtained by Lenagan and the authors through a different approach.Comment: 11 page