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Equivariant $K$-theory of regular compactifications: further developments
V. Uma Indian Institute of Technology Madras
Abstract:
We describe the
$\widetilde G\times \widetilde G$-equivariant
$K$-ring
of
$X$, where
$\widetilde G$ is a
factorial covering of a connected
complex reductive algebraic group
$G$, and
$X$ is a regular compactification
of
$G$. Furthermore, using the description
of
$K_{\widetilde G\times\widetilde G}(X)$, we describe the ordinary
$K$-ring
$K(X)$ as a free module (whose rank is equal to the cardinality
of the Weyl group) over the
$K$-ring of a toric bundle over
$G/B$ whose
fibre is equal to the toric variety
$\overline{T}^{+}$ associated with
a smooth subdivision of the positive Weyl chamber. This generalizes our
previous work on the wonderful compactification (see [1]). We also give
an explicit presentation of
$K_{\widetilde G\times\widetilde G}(X)$ and
$K(X)$
as algebras over $K_{\widetilde G\times\widetilde G}(\overline{G_{\operatorname{ad}}})$
and
$K(\overline{G_{\operatorname{ad}}})$ respectively, where
$\overline{G_{\operatorname{ad}}}$ is the wonderful compactification of the
adjoint semisimple group
$G_{\operatorname{ad}}$. In the case when
$X$ is
a regular compactification of
$G_{\operatorname{ad}}$, we give a geometric
interpretation of these presentations in terms of the equivariant and
ordinary Grothendieck rings of a canonical toric bundle
over
$\overline{G_{\operatorname{ad}}}$.
Keywords:
equivariant $K$-theory, regular compactification, wonderful compactification, toric bundle.
UDC:
512.736+
512.743
MSC: Primary
19L47; Secondary
14M25,
14M27,
14L10. Received: 28.04.2015
DOI:
10.4213/im8407
Fulltext:
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