We prove an analogue of Weyl’s law for quantized irreducible generalized flag manifolds. This is formulated in terms of a zeta function which, similarly to the classical setting, satisfies the following two properties: as a functional on the quantized algebra it is proportional to the Haar state and its first singularity coincides with the classical dimension. The relevant formulas are given for the more general case of compact quantum groups.

1.
Connes
,
A.
,
Noncommutative Geometry
(
Academic Press
,
1995
).
2.
Delius
,
G. W.
and
Gould
,
M. D.
, “
Quantum Lie algebras, their existence, uniqueness and q-antisymmetry
,”
Commun. Math. Phys.
185
(
3
),
709
-
722
(
1997
).
3.
Feger
,
R.
and
Kephart
,
T. W.
, “
LieART–A mathematica application for lie algebras and representation theory
,”
Comput. Phys. Commun.
192
,
166
-
195
(
2015
).
4.
Fuchs
,
J.
,
Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory
(
Cambridge University Press
,
1995
).
5.
Heckenberger
,
I.
and
Kolb
,
S.
, “
The locally finite part of the dual coalgebra of quantized irreducible flag manifolds
,”
Proc. London Math. Soc.
89
(
2
),
457
-
484
(
2004
).
6.
Heckenberger
,
I.
and
Kolb
,
S.
, “
De Rham complex for quantized irreducible flag manifolds
,”
J. Algebra
305
(
2
),
704
-
741
(
2006
).
7.
Helgason
,
S.
,
Differential Geometry, Lie Groups, and Symmetric Spaces
(
Academic Press
,
1979
), Vol.
80
.
8.
Klimyk
,
A. U.
and
Schmüdgen
,
K.
,
Quantum Groups and Their Representations
(
Springer
,
Berlin
,
1997
), Vol.
552
.
9.
Knapp
,
A. W.
,
Lie Groups Beyond an Introduction
(
Springer
,
2002
), Vol.
140
.
10.
Krähmer
,
U.
, “
Dirac operators on quantum flag manifolds
,”
Lett. Math. Phys.
67
(
1
),
49
-
59
(
2004
).
11.
Krähmer
,
U.
and
Tucker-Simmons
,
M.
, “
On the Dolbeault-Dirac operator of quantized symmetric spaces
,”
Trans. London Math. Soc.
2
(
1
),
33
-
56
(
2015
).
12.
Krämer
,
M.
, “
Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen
,”
Compos. Math.
38
(
2
),
129
-
153
(
1979
), https://eudml.org/doc/89398.
13.
Links
,
J. R.
and
Gould
,
M. D.
, “
Casimir invariants for Hopf algebras
,”
Rep. Math. Phys.
31
(
1
),
91
-
111
(
1992
).
14.
Matassa
,
M.
, “
Quantum dimension and quantum projective spaces
,”
Symmetry, Integrability Geom.: Methods Appl.
10
,
097
(
2014
).
15.
Matassa
,
M.
, “
Non-commutative integration, zeta functions and the Haar state for SUq(2)
,”
Math. Phys., Anal. Geom.
18
(
1
), (published online
2015
).
16.
Neshveyev
,
S.
and
Tuset
,
L.
, “
The Martin boundary of a discrete quantum group
,”
J. Reine Angew. Math.
2004
,
23
-
70
.
17.
Neshveyev
,
S.
and
Tuset
,
L.
, “
A local index formula for the quantum sphere
,”
Commun. Math. Phys.
254
(
2
),
323
-
341
(
2005
).
18.
Stokman
,
J. V.
and
Dijkhuizen
,
M. S.
, “
Quantized flag manifolds and irreducible *-representations
,”
Commun. Math. Phys.
203
(
2
),
297
-
324
(
1999
).
19.
Sugiura
,
M.
, “
Representations of compact groups realized by spherical functions on symmetric spaces
,”
Proc. Jpn. Acad.
38
(
3
),
111
-
113
(
1962
).
20.
Weyl
,
H.
, “
Über die asymptotische Verteilung der Eigenwerte
,”
Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl.
1911
,
110
-
117
, https://eudml.org/doc/58792.
21.
Zhang
,
R. B.
,
Gould
,
M. D.
, and
Bracken
,
A. J.
, “
Quantum group invariants and link polynomials
,”
Commun. Math. Phys.
137
(
1
),
13
-
27
(
1991
).
You do not currently have access to this content.