What question did this study set out to answer?

To establish explicit branching laws and direct integral decompositions for Stein’s complementary series representations and Speh representations of GL(2n,R) when restricted to the subgroup GL(2n-1,R).

August 30, 2026Algebras and Representation TheoryOpen Access

Branching Laws for Stein’s Complementary Series and Speh Representations of GL(2n,R)

Q: What does this research mean for the field?

The restriction of Stein's complementary series and Speh representations of GL(2n, R) to GL(2n-1, R) explicitly decomposes as a direct integral of unitarily induced representations from a maximal parabolic subgroup of GL(2n-1, R). Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

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To establish explicit branching laws and direct integral decompositions for Stein’s complementary series representations and Speh representations of GL(2n,R) when restricted to the subgroup GL(2n-1,R).
Analyzed the restriction of continuous representations of GL(2n,R) to GL(2n-1,R).
Constructed and evaluated two families of symmetry breaking operators acting between degenerate series representations.
Derived the explicit direct integral decomposition into unitarily induced representations from a maximal parabolic subgroup of GL(2n-1,R) with Levi factor GL(2n-2,R) x GL(1,R).
Identified that induction data comprise a complementary series or Speh representation of GL(2n-2,R) matching the original GL(2n,R) parameter alongside a character of GL(1,R).
Demonstrated compatibility of the resulting branching formulas with the theory of adduced representations.

Implication

Theoretical analysis uncovers explicit branching laws for Stein’s complementary series and Speh representations of GL(2n,R), highlighting structural alignments with adduced representations.

Abstract

We obtain the explicit direct integral decomposition of Stein’s complementary series representations and Speh representations of \,GL\,(2n,R) GL ( 2 n , R ) when restricted to the subgroup \,GL\,(2n-1,R) GL ( 2 n - 1 , R ) . The decomposition is a direct integral of unitarily induced representations from a maximal parabolic subgroup of \,GL\,(2n-1,R) GL ( 2 n - 1 , R ) with Levi factor \,GL\,(2n-2,R)× \,GL\,(1,R) GL ( 2 n - 2 , R ) × GL ( 1 , R ) , where the induction data consists of a complementary series or Speh representation of the factor \,GL\,(2n-2,R) GL ( 2 n - 2 , R ) with the same parameter as the one of \,GL\,(2n,R) GL ( 2 n , R ) and a character of \,GL\,(1,R) GL ( 1 , R ) . These results are in line with the theory of adduced representations. The main tools in the proof are two families of symmetry breaking operators between degenerate series representations of \,GL\,(2n,R) GL ( 2 n , R ) and \,GL\,(2n-1,R) GL

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Cite This Study

Ditlevsen et al. (2026) studied this question.

synapsesocial.com/papers/6a93f0576c1a8fb52e79c659 https://doi.org/10.1007/s10468-026-10413-5

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Implication

Theoretical analysis uncovers explicit branching laws for Stein’s complementary series and Speh representations of GL(2n,R), highlighting structural alignments with adduced representations.

Branching Laws for Stein’s Complementary Series and Speh Representations of GL(2n,R)

Key Points

Implication

Abstract

Discussion

Cite This Study

Branching Laws for Stein’s Complementary Series and Speh Representations of GL(2n,R)

Key Points

Implication

Abstract

Discussion

Cite This Study

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