2,528 research outputs found
Unipotent representations of real classical groups
Let be a complex orthogonal or complex symplectic group, and let
be a real form of , namely is a real orthogonal group, a
real symplectic group, a quaternionic orthogonal group, or a quaternionic
symplectic group. For a fixed parity , we
define a set of nilpotent
-orbits in (the Lie algebra of ). When
is the parity of the dimension of the standard module of , this is the set of the stably trivial special nilpotent orbits, which
includes all rigid special nilpotent orbits. For each , we construct all unipotent
representations of (or its metaplectic cover when is a real symplectic
group and is odd) attached to via the method of theta
lifting and show in particular that they are unitary
Local models of Shimura varieties, I. Geometry and combinatorics
We survey the theory of local models of Shimura varieties. In particular, we
discuss their definition and illustrate it by examples. We give an overview of
the results on their geometry and combinatorics obtained in the last 15 years.
We also exhibit their connections to other classes of algebraic varieties such
as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians
and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the
"Handbook of Moduli
Theta series and generalized special cycles on Hermitian locally symmetric manifolds
We study generalized special cycles on Hermitian locally symmetric spaces
associated to the groups ,
and . These cycles are (covered
by) locally symmetric spaces associated to subgroups of which are of the
same type. Using oscillator representation and a construction which essentially
comes from the thesis of Greg Anderson, we show that Poincar\'e duals of these
generalized special cycles can be viewed as Fourier coefficients of a theta
series. This gives new cases of theta lifts from the cohomology of Hermtian
locally symmetric manifolds associated to to vector valued automorphic
forms associated to the groups , or
which forms a reductive dual pair with in the sense of
Howe
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