[Paper Review] On the restriction of Zuckerman's derived functor modules A_q(\lambda) to reductive subgroups

Yoshiki Oshima|arXiv (Cornell University)|Jul 14, 2011

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper establishes a branching law for Zuckerman's derived functor modules $A_q(\lambda)$ restricted to reductive subgroups in symmetric pairs of real reductive Lie groups. Using D-module realizations on complex partial flag varieties, it constructs an injective homomorphism from $A_q(\lambda)$ into a direct sum of $A_{q''}(\lambda')$ modules on the subgroup, proving that each irreducible constituent is isomorphic to a submodule of such a derived functor module. The key result is the equality of the projected associated variety of $A_q(\lambda)$ and the associated variety of any irreducible constituent in the restriction, confirming a conjecture on associated varieties in the discretely decomposable case.

ABSTRACT

In this article, we study the restriction of Zuckerman's derived functor (g,K)-modules A_q(\lambda) to g' for symmetric pairs of reductive Lie algebras (g,g'). When the restriction decomposes into irreducible (g',K')-modules, we give an upper bound for the branching law. In particular, we prove that each (g',K')-module occurring in the restriction is isomorphic to a submodule of A_q'(\lambda') for a parabolic subalgebra q' of g', and determine their associated varieties. For the proof, we construct A_q(\lambda) on complex partial flag varieties by using D-modules.

Motivation & Objective

• To determine the branching laws of Zuckerman's derived functor modules $A_q(\lambda)$ when restricted to reductive subgroups in symmetric pairs.
• To establish an upper bound for the multiplicity of irreducible $(g', K')$-modules in the restriction $A_q(\lambda)|_{(g', K')}$.
• To characterize the associated varieties of irreducible constituents in the restriction, particularly verifying the equality $\mathrm{pr}_{g\to g'}(\mathrm{Ass}_g(A_q(\lambda))) = \mathrm{Ass}_{g'}(V)$ for irreducible $V$ in the decomposition.
• To extend the generalized Blattner formula to non-compact subgroups via a D-module realization of $A_q(\lambda)$ on complex partial flag varieties.

Proposed method

• Realize $A_q(\lambda)$ as the global sections of a twisted D-module on a complex partial flag variety using the framework of cohomological induction.
• Construct an injective $(g', K')$-homomorphism from $A_q(\lambda)$ into a direct sum of $A_{q''}(\lambda')$ modules on the subgroup $g'$, with explicit parabolic subalgebra $q''$ and multiplicity functions $m(\lambda', p)$.
• Use the Borel–Weil theorem and pushforward of D-modules to relate global sections on flag varieties to irreducible representations.
• Apply the Mackey isomorphism and properties of derived functors to compare $A_q(\lambda)$ with $A_q(\lambda + 2N\rho(u))$ for large $N$, ensuring compatibility with associated variety theory.
• Leverage the criterion for discrete decomposability from Kobayashi to define the relevant parabolic subalgebras $q''$ in $g'$.
• Prove the equality of projected and associated varieties by showing that $\mathrm{pr}_{g\to g'}(\mathrm{Ad}(K)(\bar{u} \cap p)) = \mathrm{Ad}(K')(u'' \cap p')$ via density and closedness arguments on flag varieties.

Experimental results

Research questions

• RQ1 Under what conditions does the restriction of $A_q(\lambda)$ to a reductive subgroup $g'$ decompose into irreducible $(g', K')$-modules?
• RQ2 What is the precise structure of the branching law for $A_q(\lambda)|_{(g', K')}$ when the restriction is discretely decomposable?
• RQ3 How are the associated varieties of irreducible constituents in the restriction related to the associated variety of $A_q(\lambda)$?
• RQ4 Can the generalized Blattner formula for compact subgroup restrictions be extended to non-compact symmetric subgroups via D-module techniques?
• RQ5 Does the equality $\mathrm{pr}_{g\to g'}(\mathrm{Ass}_g(A_q(\lambda))) = \mathrm{Ass}_{g'}(V)$ hold for irreducible $V$ in the decomposition, as conjectured in prior work?

Key findings

• The restriction $A_q(\lambda)|_{(g', K')}$ admits an injective $(g', K')$-homomorphism into a direct sum of $A_{q''}(\lambda')$ modules, with $q''$ a parabolic subalgebra of $g'$ and multiplicities $m(\lambda', p)$ explicitly defined.
• Each irreducible $(g', K')$-module occurring in the restriction is isomorphic to a submodule of some $A_{q''}(\lambda')$, generalizing the compact case formula.
• The associated variety of any irreducible constituent $V$ in the restriction satisfies $\mathrm{Ass}_{g'}(V) = \mathrm{Ad}(K')(u'' \cap p')$, where $u'' \cap p'$ is the nilradical of the parabolic $q''$.
• The projected associated variety $\mathrm{pr}_{g\to g'}(\mathrm{Ass}_g(A_q(\lambda)))$ equals $\mathrm{Ad}(K')(u'' \cap p')$, confirming the conjecture $\mathrm{pr}_{g\to g'}(\mathrm{Ass}_g(W)) = \mathrm{Ass}_{g'}(V)$ for $W = A_q(\lambda)$.
• The D-module realization of $A_q(\lambda)$ on complex partial flag varieties allows a geometric proof of the branching law via pushforward and global sections.
• The equality $\mathrm{pr}_{g\to g'}(\mathrm{Ad}(K)(\bar{u} \cap p)) = \mathrm{Ad}(K')(u'' \cap p')$ holds due to density of $K'/(Q \cap K')$ in $K/(Q \cap K)$ and closedness of the orbit closure.