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Graded Multiplicities in the Kostant-Rallis Setting

Andrew Frohmader

TL;DR

This work advances the Kostant–Rallis program by providing explicit, stable branching rules for GL_n down to O_n and GL_{2n} down to Sp_{2n}, extending classical Littlewood restriction rules to all parameters. It then derives a combinatorial, crystal-based formula for the graded multiplicities of K-types in the regular functions on the K-nilpotent cone, linking these polynomials to the Hodge K-character of spherical principal series with infinitesimal character 0. The approach furnishes a concrete, tableau-driven description via GL_n crystals and seesaw-type LR rules, enabling explicit computation of m^{\nu,0}_{(G,K)}(q) in the key cases and connecting to invariant theory and Hodge-theoretic interpretations. The results generalize known Kostant-exponent formulas, provide new combinatorial tools (generalized LR coefficients and $M$-polynomials), and illuminate the unitarity/Hodge filtration picture for spherical principal series in the Kostant–Rallis setting. A broad potential extension to other symmetric pairs and non-spherical principal series is highlighted, with a crystal-theoretic mechanism at the core of the construction.

Abstract

This paper contains two main results. First, we provide combinatorial branching rules for $\text{GL}_n \downarrow \text{O}_n$ and $\text{GL}_{2n} \downarrow \text{Sp}_{2n}$ extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of $\text{GL}_n$-crystals to derive a formula for the graded multiplicity of a $K$-type in the regular functions on the $K$-nilpotent cone for $\text{GL}(n, \mathbb{R})$, $\text{GL}(n, \mathbb{C})$ and $\text{GL}(n, \mathbb{H})$. Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge $K$-character of the spherical principal series with infinitesimal character 0.

Graded Multiplicities in the Kostant-Rallis Setting

TL;DR

This work advances the Kostant–Rallis program by providing explicit, stable branching rules for GL_n down to O_n and GL_{2n} down to Sp_{2n}, extending classical Littlewood restriction rules to all parameters. It then derives a combinatorial, crystal-based formula for the graded multiplicities of K-types in the regular functions on the K-nilpotent cone, linking these polynomials to the Hodge K-character of spherical principal series with infinitesimal character 0. The approach furnishes a concrete, tableau-driven description via GL_n crystals and seesaw-type LR rules, enabling explicit computation of m^{\nu,0}_{(G,K)}(q) in the key cases and connecting to invariant theory and Hodge-theoretic interpretations. The results generalize known Kostant-exponent formulas, provide new combinatorial tools (generalized LR coefficients and -polynomials), and illuminate the unitarity/Hodge filtration picture for spherical principal series in the Kostant–Rallis setting. A broad potential extension to other symmetric pairs and non-spherical principal series is highlighted, with a crystal-theoretic mechanism at the core of the construction.

Abstract

This paper contains two main results. First, we provide combinatorial branching rules for and extending the Littlewood restriction rules. Second, we use these branching rules and the combinatorics of -crystals to derive a formula for the graded multiplicity of a -type in the regular functions on the -nilpotent cone for , and . Due to work of Schmid and Vilonen, these graded multiplicities determine the Hodge -character of the spherical principal series with infinitesimal character 0.
Paper Structure (28 sections, 55 theorems, 126 equations)

This paper contains 28 sections, 55 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Representation theory preliminaries
  3. Reductive Lie algebras
  4. Conventions for $\mathop{\mathrm{GL}}\nolimits_n$
  5. Parameterization of representations and $M$-polynomials
  6. The ring of invariants $\mathop{\mathrm{\mathbb{C}}}\nolimits[\mathop{\mathrm{\mathfrak{p}}}\nolimits]^K$
  7. $\mathop{\mathrm{\mathbb{C}}}\nolimits[\mathop{\mathrm{\mathfrak{p}}}\nolimits]$ as a multiplicity-free space
  8. Crystals of reductive Lie algebras
  9. Definitions
  10. Tensor products of crystals
  11. Normal crystals
  12. Crystal anti-isomorphisms
  13. $\mathop{\mathrm{GL}}\nolimits_n$-crystals and tableaux
  14. Tableaux and Knuth equivalence
  15. $\mathbb{B}_n^{\otimes k}$ and the signature rule
  16. ...and 13 more sections

Key Result

Theorem 1.1

KostantRallis1971 The ideal $\mathop{\mathrm{\mathbb{C}}}\nolimits[\mathop{\mathrm{\mathfrak{p}}}\nolimits]^K_+$ is the ideal associated to the variety $\mathcal{N}_\theta$, i.e. $\mathop{\mathrm{\mathbb{C}}}\nolimits[\mathop{\mathrm{\mathfrak{p}}}\nolimits]^K_+ = \{f \in \mathop{\mathrm{\mathbb{C}}

Theorems & Definitions (127)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['thm:main_branching']} below
  • Theorem 1.7
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • ...and 117 more