arXiv subjects

Alexander Alldridge

Publications and source records attributed to Alexander Alldridge.

21 records · Page 2Linked to original sources

arXiv2009

Chevalley's restriction theorem for reductive symmetric superpairs

Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k+k,k) with the flip involution where k is a classical Lie superalgebra with a non-degenerate invariant even form (equivalently, a finite-dimensional contragredient Lie superalgebra). Thus, we obtain a new proof of the generalisation of Chevalley's restriction theorem due to Sergeev and Kac, Gorelik. For general symmetric superpairs, the invariants exhibit a new and surprising behaviour. We illustrate this phenomenon by a detailed discussion in the example g=C(q+1)=osp(2|2q,C), endowed with a special involution. In this case, the invariant algebra defines a singular algebraic curve.

math.RT↗

arXiv2009

Spectrum and Analytical Indices of the C*-Algebra of Wiener-Hopf Operators

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener--Hopf C$^*$-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in $K$-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.

math.OA↗

arXiv2009

Boundary orbit strata and faces of invariant cones and complex Olshanskii semigroups

Let D=G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification, and there exists a closed complex subsemigroup, the so-called minimal Olshanskii semigroup, of the complexification characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to it. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Koranyi and Wolf, we give a detailed and complete description of the K-orbit type strata of the minimal Olshanskii semigroup, as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra of G.

math.CV↗