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On extended weight monoids of spherical homogeneous spaces. (English) Zbl 1474.14069

Let \(G\) be a connected reductive algebraic group, let \(B\) be a Borel subgroup of \(G\) and \(H\) a spherical subgroup of \(G\) – i.e. \(B\) acts with an open orbit on the quotient \(G/H\). Let \(\mathfrak{X}(B)\) and \(\mathfrak{X}(H)\) be the group of characters of \(B\) and \(H\) respectively, and let \(\Lambda^+(G)\subset \mathfrak{X}(B)\) be the monoid of dominant weights with respect to \(B\). For \(\lambda\in\Lambda^+(G)\), let \(V_G(\lambda)^*\) be the dual of the irreducible representation of \(G\) of dominant weight \(\lambda\). The extended monoid of the quotient \(G/H\) is defined by \[ \widehat{\Lambda}^+(G/H)= \{(\lambda,\chi )\in\Lambda^+(G)\times\mathfrak{X}(H)\ |\ [V_G(\lambda)^* ]^{(H)}_\chi\neq 0 \} \] where \([V_G(\lambda)^* ]^{(H)}_\chi\) is the \(H\)-semi-invariant of weight \(\chi\) in \(V_G(\lambda)^*\).
This extended monoid is an interesting invariant of the spherical space \(G/H\), indeed the invariants \(\mathcal{D}_G(G/H)\) – the color of \(G/H\) i.e. \(B\) stable prime divisor of \(G/H\) – and \(\Lambda(G/H)\) – the weights of \(G/H\) – can be expressed in term of \(\widehat{\Lambda}^+(G/H)\). In this paper the author gives a description of the extended monoid \(\widehat{\Lambda}^+(G/H)\) when \(G\) semi-simple, simply connected and \(H\) is specified by a regular embedding in a parabolic subgroup \(P\) of \(G\). The practical description of the extended monoid depends on the knowledge of the set of simple spherical root of \(G/H\). The author gives some tools which make it possible to calculate these spherical roots in some cases, notably in the case where \(H\) is strongly solvable, which allows him to give a new proof of a previous result of R. S. Avdeev and N. E. Gorfinkel [Funct. Anal. Appl. 46, No. 3, 161–172 (2012; Zbl 1279.43011); translation from Funkts. Anal. Prilozh. 46, No. 3, 1–15 (2012)]. The paper ends with complete calculations of the generators of the extended monoid in three explicit examples.

MSC:

14M17 Homogeneous spaces and generalizations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 1279.43011
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References:

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