buildings of spherical type
Andries E. Brouwer
Department of Mathematics, Technological University Eindhoven,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
To Reza Khosrovshahi on the occasion of his 70th birthday
Consider the graph Γ obtained by taking as vertices the flags in a ﬁnite
building of spherical type deﬁned over Fq, where two flags are adjacent
when they are opposite. We show that the squares of the eigenvalues of
Γ are powers of q.
Let G be a ﬁnite group of Lie type with Borel subgroup B and Weyl group
W , so that one has the Bruhat decomposition G =
BwB. Let (W, S) be
a Coxeter system, and let w0 be the longest element of W w.r.t. the set of
generators S. Then conjugation by w0 induces a diagram automorphism on the
Coxeter diagram of W (with vertex set S).
Let a type be a nonempty subset of S. Call two types J, K opposite when
K = J w0 (so that J = Kw0 ).
For J ⊂ S, let WJ := J and PJ := BWJ B. Let an object of type S \ J, or
of cotype J, be a coset gPJ in G. Call two objects gPJ and hPK opposite when
their cotypes J, K are opposite, and moreover PK
= PK w0PJ .
Let ΓJ,K , with K = J
, be the bipartite graph with as vertices in one part
the objects of cotype J and in the other part the objects of cotype K, where two
vertices in diﬀerent parts are adjacent when they are opposite. If J = K, let ΓJ
be the graph with as vertices the objects of cotype J, adjacent when opposite.
Theorem 1.1 Let G be deﬁned over Fq. Let J be a proper subset of S, and let
K = J
. Let θ be an eigenvalue of ΓJ,K or, if J = K, of ΓJ . Then
for some integer e.
The exponents e can be determined explicitly.
2010 Math Subject Classiﬁcation: 05C50, 05Exx, 20E42, 20G40, 51N30
Volume 531, 2010