Square integrable highest weight representations

Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

The oscillator method for the construction of unitary highest (or lowest) weight representations of noncompact groups and supergroups is generalized. Within this generalization, the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group. The method is illustrated by studying in detail the unitary highest weight representations of the supergroup OSp (2n+1/2m,R). The generalized supercoherent states associated with these unitary representations are also defined.

Let g be a semisimple Lie algebra with h a Cartan subalgebra. The orbit method attempts to assign representations of g to orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits which should lead to highest weight representations of g. It is known that all unitary highest weight representations can be obtained by their quantization [8]. A hypersurface orbital variety is one which is of codimension 1 in the nilradical of a parabolic. Their classification for g = sl(n) is obtained from general results in [12]. Recently Benlolo and Sanderson [2] conjectured the form of the (nonlinear) element describing such a variety. This paper proves that conjecture and further constructs a simple module with integral highest weight which “quantizes” the variety in the precise sense that the regular functions of its closure is given a g module structure compatible (up to shift by the highest weight) with its h module structure.KeywordsWord FormYoung TableauNilpotent OrbitHigh Weight VectorHigh Weight ModuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Gradient-type differential operators and unitary highest weight representations of SU( p, q)

We use Kirillov's orbit method to construct all unitarizable highest weight representations with scalar lowest K-type.

The quotient $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ of a super loop group $$\widetilde{LG}$$ by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding with a coadjoint orbit of the centrally extended super loop algebra . The algebra $$\widetilde{diff}^c S^1 $$ of super-conformal vector fields on the circle is shown to have a natural representation as Hamiltonian vector fields on $${{\widetilde{LG}} \mathord{\left/ {\vphantom {{\widetilde{LG}} G}} \right. \kern-\nulldelimiterspace} G}$$ generated by an equivariant moment map . This map is obtained by composition of315-8 with a super Poisson map defining a supersymmetric extension of the classical Sugawara formula. Upon quantization, this yields the corresponding formula of Kac and Todorov on unitary highest weight representations. For any homomorphism ρ:u(1)→G, an associated “twisted” moment map is also derived, generating a super Poisson bracket realization of a super Virasoro subalgebra $$\widetilde{Vir}$$ of the semi-direct sum . The corresponding super Poisson map is interpreted as a nonabelian generalization of the super Miura map and applied to two super KdV hierarchies to derive corresponding integrable generalized super MKdV hierarchies in .

Let n≥2 be a positive integer. To each irreducible representation σ of Sp(1), an Sp(1)-Kepler problem in dimension (4n−3) is constructed and analyzed. This system is superintegrable, and when n=2 it is equivalent to a generalized MICZ-Kepler problem in dimension of 5. The dynamical symmetry group of this system is Õ∗(4n) with the Hilbert space of bound states H(σ) being the unitary highest weight representation of O∗˜(4n) with highest weight, (−1,⋯,−1,−(1+σ¯)), which occurs at the rightmost nontrivial reduction point in the Enright–Howe–Wallach classification diagram for the unitary highest weight modules. Here σ¯ is the highest weight of σ. Furthermore, it is shown that the correspondence σ↔H(σ) is the theta-correspondence for dual pair (Sp(1),O∗(4n))⊆Sp(8n,R).

We determine precisely for which irreducible unitary highest weight representation of the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the countable direct limit of the finite-dimensional unitary groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the corresponding 1-cohomology space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> does not vanish. This occurs in particular if a highest weight, viewed as an integer-valued function on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℕ</mml:mi> </mml:math> , is finitely supported. In a second step, we extend the finitely supported highest weight representations to norm-continuous unitary representations of the Banach-completions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the direct limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> th Schatten norm for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the corresponding 1-cohomology spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> do not vanish either, except in three cases. We conclude that these groups do not have Kazhdan’s Property (T). On the other hand, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the first cohomology spaces all vanish because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has property (FH) as a bounded topological group.

Let Ω be a bounded circular domain in ℂ N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles.

The branching rules for unitary highest weight representations of Virasoro and super-Virasoro algebras that contain a subalgebra with central charge c=1 are presented. These rules are useful for identifying points of higher symmetry in two-dimensional critical systems with c=1 and for the analysis of a defected Ising chain.

For each integer $n\ge 2$, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem has an $\widetilde{\mr{Spin}}(2, 2n+1)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module.

For a local non-Archimedean field K K we construct G L d + 1 ( K ) {\mathrm {GL}}_{d+1}(K) -equivariant coherent sheaves V O K {\mathcal V}_{{\mathcal O}_K} on the formal O K {\mathcal O}_K -scheme X {\mathfrak X} underlying the symmetric space X X over K K of dimension d d . These V O K {\mathcal V}_{{\mathcal O}_K} are O K {\mathcal O}_K -lattices in (the sheaf version of) the holomorphic discrete series representations (in K K -vector spaces) of G L d + 1 ( K ) {\mathrm {GL}}_{d+1}(K) as defined by P. Schneider. We prove that the cohomology H t ( X , V O K ) H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K}) vanishes for t &gt; 0 t&gt;0 , for V O K {\mathcal V}_{{\mathcal O}_K} in a certain subclass. The proof is related to the other main topic of this paper: over a finite field k k , the study of the cohomology of vector bundles on the natural normal crossings compactification Y Y of the Deligne-Lusztig variety Y 0 Y^0 for G L d + 1 / k {\mathrm {GL}}_{d+1}/k (so Y 0 Y^0 is the open subscheme of P k d {\mathbb P}_k^d obtained by deleting all its k k -rational hyperplanes).

Let \pi be a cuspidal automorphic representation of PGL _2(\mathbb A_\mathbb Q) of arithmetic conductor C and archimedean parameter T , and let \phi be an L^2 -normalized automorphic form in the space of \pi . The sup-norm problem asks for bounds on \| \phi \|_\infty in terms of C and T . The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the L^2 -mass |\phi|^2 (g) \, d g of \phi . Previous work on these problems in the conductor-aspect has focused on the case that \phi is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor p of C , the local component \pi_p is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms \phi for which the local components \phi_p \in \pi_p are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of PGL _2(\mathbb{R}) . For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if \pi_\infty is a holomorphic discrete series of lowest weight k , we obtain the optimal bound C^{1/8 -\epsilon} k^{1/4 - \epsilon} \ll_{\epsilon} |\phi|_\infty \ll_{\epsilon} C^{1/8 + \epsilon} k^{1/4+\epsilon} . We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence [31] for newforms of powerful level.

We prove uniqueness and give precise criteria for existence of split and nonsplit Bessel models for a class of lowest and highest weight representations of the groups and including all holomorphic and antiholomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global L-function on GSp(4) × GL(2) involving a vector-valued Siegel modular form of degree 2.

Characters of some unitary highest weight representations via the theta correspondence