Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$

DOI10.1007/S00209-022-03199-XarXiv2211.10664MaRDI QIDQ6417783

Dmitri Panyushev, Oksana S. Yakimova

Publication date: 19 November 2022

Abstract: Let

m a t h f r a k g

be a semisimple Lie algebra,

v a r t h e t a i n s f A u t (m a t h f r a k g)

a finite order automorphism, and

m a t h f r a k g_{0}

the subalgebra of fixed points of

v a r t h e t a

. Recently, we noticed that using

v a r t h e t a

one can construct a pencil of compatible Poisson brackets on

m a t h c a l S (m a t h f r a k g)

, and thereby a `large' Poisson-commutative subalgebra

m a t h c a l Z (m a t h f r a k g, v a r t h e t a)

of

m a t h c a l S (m a t h f r a k g)^{m a t h f r a k g_{0}}

. In this article, we study invariant-theoretic properties of

(m a t h f r a k g, v a r t h e t a)

that ensure good properties of

m a t h c a l Z (m a t h f r a k g, v a r t h e t a)

. Associated with

v a r t h e t a

one has a natural Lie algebra contraction

m a t h f r a k g_{(0)}

of

m a t h f r a k g

and the notion of a good generating system (=g.g.s.) in

m a t h c a l S (m a t h f r a k g)^{m} a t h f r a k g

. We prove that in many cases the equality

m a t h s f i n d, m a t h f r a k g_{(0)} = m a t h s f i n d, m a t h f r a k g

holds and

m a t h c a l S (m a t h f r a k g)^{m} a t h f r a k g

has a g.g.s. According to V.G. Kac's classification of finite order automorphisms (1969),

v a r t h e t a

can be represented by a Kac diagram,

m a t h c a l K (v a r t h e t a)

, and our results often use this presentation. The most surprising observation is that

m a t h f r a k g_{(0)}

depends only on the set of nodes in

m a t h c a l K (v a r t h e t a)

with nonzero labels, and that if

v a r t h e t a

is inner and a certain label is nonzero, then

m a t h f r a k g_{(0)}

is isomorphic to a parabolic contraction of

m a t h f r a k g

.

Mathematics Subject Classification ID

Group actions on varieties or schemes (quotients) (14L30) Semisimple Lie groups and their representations (22E46) Poisson algebras (17B63) Simple, semisimple, reductive (super)algebras (17B20) Coadjoint orbits; nilpotent varieties (17B08)

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