Invariant differential operators on spherical homogeneous spaces with overgroups

zbMATH Open1430.16027arXiv1810.02803MaRDI QIDQ5241595

Author name not available (Why is that?)

Publication date: 31 October 2019

Abstract: We investigate the structure of the ring

{m a t h b b D}_{G} (X)

of

G

-invariant differential operators on a reductive spherical homogeneous space

X = G / H

with an overgroup

w i d e t i l d e G

. We consider three natural subalgebras of

{m a t h b b D}_{G} (X)

which are polynomial algebras with explicit generators, namely the subalgebra

{m a t h b b D}_{w i d e t i l d e G} (X)

of

w i d e t i l d e G

-invariant differential operators on

X

and two other subalgebras coming from the centers of the enveloping algebras of

m a t h f r a k g

and

m a t h f r a k k

, where

K

is a maximal proper subgroup of

G

containing

H

. We show that in most cases

{m a t h b b D}_{G} (X)

is generated by any two of these three subalgebras, and analyze when this may fail. Moreover, we find explicit relations among the generators for each possible triple

(w i d e t i l d e G, G, H)

, and describe "transfer maps" connecting eigenvalues for

{m a t h b b D}_{w i d e t i l d e G} (X)

and for the center

Z ({m a t h f r a k g}_{m a t h b b C})

of the enveloping algebra of

{m a t h b b g}_{m a t h b b C}

.

Full work available at URL: https://arxiv.org/abs/1810.02803

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