82 research outputs found
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
reduction of the Springer correspondent representation involves the sign
representation exactly when 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 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
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