## Asian Journal of Mathematics

- Asian J. Math.
- Volume 8, Number 4 (2004), 685-732.

### The Chevallry Involution and a Duality of Weight Varieties

Bejamin J. Howard and John J. Millson

#### Abstract

In this paper we show that the classical notion of association of projective
point sets,[DO], Chapter III, is a special case of a general duality
between weight varieties (i.e. torus quotients of flag manifolds) of a
reductive group *G* induced by the action of the Chevalley involution
on the set of these quotients. We compute the dualities explicitly on both
the classical and quantum levels for the case of the weight varieties
associated to *GL*_{n}(**C**). In particular we obtain the
following formula as a special case. Let **r** = (*r*_{1},
...,*r*_{n})be an *n*-tuple of positive real numbers and
*M*_{r}(**CP**^{m})be the moduli
space of semistable weighted(by **r**)configurations of *n* points
in **CP**^{m} modulo projective equivalence, see for
example[FM]. Let **LAMBDA** be the vector in **R**^{n}
with all components equal to [Sigma]_{i}*r*_{
i}/(*m* + 1). Then *M*_{r}(**CP**
^{m}) *is isomorphic to* *M*_{LAMBDA-rr}
(**CP**^{n-m-2}) (the meaning of *is isomorphic to* depends on r and
will be explained below, see Theorem 1.6). We conclude by studying
"self-duality" i.e. those cases where the duality isomorphism carries the
torus quotient into itself. We characterize when such a self-duality is
trivial, i.e. equal to the identity map. In particular we show that all
self-dualities are nontrival for the weight varieties associated to the
exceptional groups. The quantum version of this problem, i.e. determining for
which self-adjoint representations *V* of *G* the Chevalley
involution acts as a scalar on the zero weight space *V*[0], is important
in connection with the irreducibility of the representations of Artin groups
of Lie type which are obtained as the monodromy of the Casimir connection, see
[MTL], and will be treated in [HMTL].

#### Article information

**Source**

Asian J. Math., Volume 8, Number 4 (2004), 685-732.

**Dates**

First available in Project Euclid: 13 June 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1118669696

**Mathematical Reviews number (MathSciNet)**

MR2127944

**Zentralblatt MATH identifier**

1093.14067

#### Citation

Howard, Bejamin J.; Millson, John J. The Chevallry Involution and a Duality of Weight Varieties. Asian J. Math. 8 (2004), no. 4, 685--732. https://projecteuclid.org/euclid.ajm/1118669696