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Special unipotent representations of real classical groups: counting and reduction

Dan Barbasch, Jia-Jun Ma, Binyong Sun, Chen-Bo Zhu

TL;DR

This work develops a framework to count and construct special unipotent representations of real classical groups by exploiting coherent continuation representations and endoscopic techniques. It establishes a counting inequality for representations with a fixed infinitesimal character and a bound on the complex associated variety, and under technical hypotheses proves an exact count in many real-classical cases. A key innovation is the reduction to the analytically even (good parity) case and a detailed combinatorial model using painted bipartitions to parameterize Unip$_{\breve{\mathcal O}}(G)$; for complex classical groups the counting reduces to a canonical $2^{\sharp(\text{PP}_\star(\breve{\mathcal O}_{\mathrm g}))}$ rule. The results unify highest-weight and Casselman–Wallach pictures via coherent continuation, provide explicit counting formulas by type (GL, U(p,q), B/C/D families), and lay groundwork for explicit construction (to be completed in the follow-up paper) of all special unipotent representations through endoscopy and theta lifting, highlighting the role of $\tau$-invariants, Lusztig cells, and dualities in guiding the classification.

Abstract

Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a given bound of the complex associated variety. When $G$ is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of $G$ attached to $\check{\mathcal O}$, in the sense of Arthur and Barbasch-Vogan. Here $\check{\mathcal O}$ is a nilpotent adjoint orbit in the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We give a precise count for the number of special unipotent representations of $G$ attached to $\check{ \mathcal O}$. We also reduce the problem of constructing special unipotent representations attached to $\check{\mathcal O}$ to the case when $\check{\mathcal O}$ is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups.

Special unipotent representations of real classical groups: counting and reduction

TL;DR

This work develops a framework to count and construct special unipotent representations of real classical groups by exploiting coherent continuation representations and endoscopic techniques. It establishes a counting inequality for representations with a fixed infinitesimal character and a bound on the complex associated variety, and under technical hypotheses proves an exact count in many real-classical cases. A key innovation is the reduction to the analytically even (good parity) case and a detailed combinatorial model using painted bipartitions to parameterize Unip; for complex classical groups the counting reduces to a canonical rule. The results unify highest-weight and Casselman–Wallach pictures via coherent continuation, provide explicit counting formulas by type (GL, U(p,q), B/C/D families), and lay groundwork for explicit construction (to be completed in the follow-up paper) of all special unipotent representations through endoscopy and theta lifting, highlighting the role of -invariants, Lusztig cells, and dualities in guiding the classification.

Abstract

Let be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of with a given infinitesimal character and a given bound of the complex associated variety. When is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of attached to , in the sense of Arthur and Barbasch-Vogan. Here is a nilpotent adjoint orbit in the Langlands dual of (or the metaplectic dual of when is a real metaplectic group). We give a precise count for the number of special unipotent representations of attached to . We also reduce the problem of constructing special unipotent representations attached to to the case when is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups.
Paper Structure (56 sections, 92 theorems, 479 equations)

This paper contains 56 sections, 92 theorems, 479 equations.

Table of Contents

  1. Introduction
  2. Background and goals
  3. Approach
  4. Organization
  5. The main results
  6. Lie algebra notation
  7. Counting irreducible representations
  8. Special unipotent representations of real classical groups
  9. General linear groups
  10. Unitary groups
  11. Orthogonal and symplectic groups: relevant parabolic subgroups
  12. Orthogonal and symplectic groups: reduction to good parity
  13. Orthogonal and symplectic groups: the case of good parity
  14. The case of complex classical groups
  15. Generalities on coherent families of highest weight modules
  16. ...and 41 more sections

Key Result

Theorem 2.1

We have the inequality where $1_{W_{\nu}}$ denotes the trivial representation of the stabilizer ${W_{\nu}}$ of $\nu$ in $W$. The equality holds if the Coxeter group $W(\Lambda)$ has no simple factor of type $F_4$, $E_6$, $E_7$, or $E_8$, and $G$ is linear or isomorphic to a real metaplectic group.

Theorems & Definitions (196)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Example 2.10
  • Definition 2.11
  • ...and 186 more