The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups

TY - JOUR

T1 - The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups

AU - Möllers, Jan

AU - Ørsted, Bent

PY - 2019

Y1 - 2019

N2 - We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a reductive Lie group G and a reductive subgroup G', and between their composition factors. Our method describes the restriction of these operators to the K'-isotypic components, K' ⊆ G' a maximal compact subgroup, and reduces the representation-theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank-one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds; i.e., every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman-Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetrybreaking operators for orthogonal groups by Kobayashi and Speh, gives new proofs of their main results, and extends them to unitary groups.

AB - We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a reductive Lie group G and a reductive subgroup G', and between their composition factors. Our method describes the restriction of these operators to the K'-isotypic components, K' ⊆ G' a maximal compact subgroup, and reduces the representation-theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank-one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds; i.e., every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman-Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetrybreaking operators for orthogonal groups by Kobayashi and Speh, gives new proofs of their main results, and extends them to unitary groups.

KW - Harish-chandra modules

KW - Intertwining operators

KW - Principal series

KW - Spectrum-generating operator

KW - Symmetry-breaking operators

U2 - 10.2140/pjm.2019.302.23

DO - 10.2140/pjm.2019.302.23

M3 - Journal article

SN - 0030-8730

VL - 302

SP - 23

EP - 76

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 1

ER -