Kyo Nishiyama

Theta lifting of unitary lowest weight modules and their associated cycles

We consider a reductive dual pair (G,G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study theta lifting of (holomorphic) nilpotent K ′ C -orbits in relation to theta lifting of unitary lowest weight modules of G′. We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under theta lifting.

Theta lifting of holomorphic discrete series: The case of (,)×(,)

Let (G,G′) = (U(n, n), U(p, q)) (p + q ≤ n) be a reductive dual pair in the stable range. We investigate theta lifts to G of unitary characters and holomorphic discrete series representations of G′, in relation to the geometry of nilpotent orbits. We give explicit formulas for their K-type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme K′-type, we show that the K structure of the resulting representations of G is almost identical to the KC-module structure of the regular function rings on the closure of the associated nilpotent KC-orbits in s, where g = k⊕ s is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

Classification of irreducible super-unitary representations ofsu(p,q/n)

In this paper we classify all the irreducible super-unitary representations ofsu(p,q/n), which can be integrated up to a unitary representation ofS(U(p,q)×U(n)), a Lie group corresponding to the even part ofsu(p,q/n). Note that a real form of the Lie superalgebrasl(m/n;ℂ) which has non-trivial superunitary representations is of the formsu(p,q/n)(p+q=m) orsu(m/r,s)(r+s=n). Moreover, we give an explicit realization for each irreducible super-unitary representation, using the oscillator representation of an orthosymplectic Lie superalgebra.

Regular Orbits of Symmetric Subgroups on Partial Flag Varieties

We give a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of Spaltenstein varieties and associated nilpotent orbits. We explain applications to enumerating special unipotent representations of real reductive groups, as well as (a portion of) the closure order on the set of nilpotent coadjoint orbits.

Algebraic Structures on Virtual Characters of a Semisimple Lie Group

Publisher Summary This chapter discusses the algebraic structures on virtual characters of a semisimple lie group. All the irreducible admissible representations of G are classified under infinitesimal equivalence. There are three different methods to classify them: Langlands classification, classification using the theory of D-modules, and Vogans minimal K-type arguments. The chapter explains that Langlands parameter of a finite dimensional representation is very difficult to calculate out. Therefore, the classification theory is not only needed but also easier description or structures of admissible representations. There are many improvements in this direction, for example, theory of primitive ideals, Kc-orbits on flag varieties, and Weyl group representations on virtual character modules. Algebraic structures are treated on virtual character modules. G. J. Zuckerman defined a representation of Weyl groups on virtual character modules on G with regular infinitesimal character. This Weyl group representation provides powerful methods to calculate invariants of admissible representations, such as Gelfand–Kirillov dimensions, τ-invariants, primitive ideals, and to classify them in large.

Asymptotic cone of semisimple orbits for symmetric pairs

Abstract Let G be a reductive algebraic group over C and denote its Lie algebra by g . Let O h be a closed G-orbit through a semisimple element h ∈ g . By a result of Borho and Kraft (1979) [4] , it is known that the asymptotic cone of the orbit O h is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer Z G ( h ) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair. More precisely, let θ be an involution of G, and K = G θ a fixed point subgroup of θ. Then we have a Cartan decomposition g = k + s of the Lie algebra g = Lie ( G ) which is the eigenspace decomposition of θ on g . Let { x , h , y } be a normal sl 2 triple, where x , y ∈ s are nilpotent, and h ∈ k semisimple. In addition, we assume x ¯ = y , where x ¯ denotes the complex conjugation which commutes with θ. Then a = − 1 ( x − y ) is a semisimple element in s , and we can consider a semisimple orbit Ad ( K ) a in s , which is closed. Our main result asserts that the asymptotic cone of Ad ( K ) a in s coincides with Ad ( G ) x ∩ s ¯ , if x is even nilpotent.

Codimension one connectedness of the graph of associated varieties

Let

Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian

\pi

A note on the Capelli identities for symmetric pairs of Hermitian type

be an irreducible Harish-Chandra

Infinite dimensional harmonic analysis IV: On the interplay between representation theory, random matrices, special functions, and probability

(\mathfrak{g}, K)