AGU RT Seminar:
Workshop on Harmonic Analysis on Lie groups andRepresentations of Hecke Algebras
Date:2023 年 3 月 20 日 (月) March 20th, Monday, 2023

Place:

青山学院大学理工学部 相模原キャンパス L603 室 (L 棟 6 階)
〒 252-5258 相模原市 中央区 淵野辺 5-10-1
Room L603, Department of Mathematics, Aoyama Gakuin UniversitySpeakers:

Pavle Pandžić (Zagreb Univ.), 大島芳樹 (東京大学)
織田寛（拓殖大学）, 久保利久（龍谷大学）, 田内大渡（九州大学）

Organizer: 西山享（青山学院大学）, Kyo Nishiyama (AGU)

3/20, Mon, 9:30–10:00: Opening, discussions
3/20, Mon, 10:00–10:50: Toshihisa Kubo (Ryukoku Univ.)
The classiﬁcation of the K-type formulas for the Heisenberg ultrahyperbolic
equation of sl(3, R)
3/20, Mon, 11:10–12:00: Pavle Pandžić (Zagreb Univ.)
Dirac index and associated cycles of Harish-Chandra modules
3/20, Mon, 13:30–14:20: Yoshiki Oshima (Univ. Tokyo)
Discrete branching laws of derived functor modules
3/20, Mon, 14:40–15:25: Hiroshi Oda (Takushoku Univ.)
Inversion formula for Opdam–Cherednik transform associated with a root systemof type BC
3/20, Mon, 15:45–16:30: Hiroshi Oda (Takushoku Univ.)
Functors connecting real reductive groups and graded Hecke algebras
3/20, Mon, 16:50–17:40: Taito Tauchi (Kyushu Univ.)
A generalization of the Kobayashi–Oshima uniformly bounded multiplicity
theorem
3/20, Mon, 17:40–18:00: Closing, discussions

Abstract
Toshihisa Kubo (Ryukoku Univ.)[3/20, Mon, 10:00–10:50]

Title: The classiﬁcation of the K-type formulas for the Heisenberg ultrahyperbolic equationof sl(3, R)(n)About ten years ago, Kable constructed a one-parameter family □s (s ∈ C)(n)of diﬀerential operators for sl(n, C) with n ≥ 3. He referred to □s as theHeisenberg ultrahyperbolic operator. In the viewpoint of intertwining operators,(n)□s can be thought of as an intertwining diﬀerential operator between certainfparabolically induced representations for SL(n,R). In this talk we discuss about
the classiﬁcation of the K-type formulas of the space of K-ﬁnite solutions to the(3)f R) and some related topics.
diﬀerential equation □s f = 0 for SL(3,This is joint work with Bent Ørsted.

Pavle Pandžić (Zagreb Univ.)[3/20, Mon, 11:10–12:00]

Title: Dirac index and associated cycles of Harish-Chandra modules
We review the construction of Dirac operators, cohomology and index for
Harish-Chandra modules, as well as associated varieties and associated cycles.
Then we show how, for certain Harish-Chandra modules, the polynomial giving
the dimension of the Dirac index of the corresponding coherent family can be expressed as an integer linear combination of the multiplicities in the characteristic
cycle. This is joint work with S.Mehdi, D.Vogan and R.Zierau.
Yoshiki Oshima (Univ. Tokyo)[3/20, Mon, 13:30–14:20]

Title: Discrete branching laws of derived functor modules
We consider the restriction of Zuckerman’s derived functor modules for symmetric pairs of real reductive groups assuming that it is discretely decomposable
in the sense of Kobayashi. By a case-by-case argument, it can be shown that therestriction decomposes as a direct sum of Zuckerman’s derived functor modules
for the subgroup. In this talk, we would like to discuss how to obtain explicit
branching formulas for some examples.

Hiroshi Oda (Takushoku Univ.)[3/20, Mon, 14:40–15:25]

Title: Inversion formula for Opdam–Cherednik transform associated with a root systemof type BC
We give the inversion formula and the Plancherel formula for Opdam–Cherednik transform associated with a root system of type BC, when the multiplicity parameters are not necessarily nonnegative. This is a non-symmetric
generalization of a recent result (DOI: 10.2969/jmsj/88728872) obtained with
N. Shimeno and T. Honda.
Tatsuo HONDA, Hiroshi ODA, Nobukazu SHIMENO ”Inversion formula forthe hypergeometric Fourier transform associated with a root system of typeBC,” Journal of the Mathematical Society of Japan, J. Math. Soc. Japan
Advance Publication, 1-37, (December, 2022)

Hiroshi Oda (Takushoku Univ.)[3/20, Mon, 15:45–16:30]

Title: Functors connecting real reductive groups and graded Hecke algebrasWith a given reductive Lie group G there is an associated graded Hecke algebra
H. In order to connect representations of G and H we introduce the notion ofradial pairs and give some examples. We also deﬁne a functor Ξ : Rep H →Rep G such that (Ξ(Y ), Y ) is a radial pair for Y ∈ Rep H. When G is insome series of classical groups, Ciubotaru and Trapa constructed a functor FCT :Rep G → Rep H. In this case, Ξ is a right inverse of FCT and any radial pair
(X, Y ) satisﬁes FCT (X) = Y .

Taito Tauchi (Kyushu Univ.)[3/20, Mon, 16:50–17:40]

Title: A generalization of the Kobayashi–Oshima uniformly bounded multiplicity theorem
Let H be a real reductive algebraic subgroup of a real reductive algebraic group
G and GC , HC their complexiﬁcations. It was proved by T. Kobayashi and T.
Oshima that the multiplicities of irreducible representations of G in the regular
representation C ∞ (G/H) are uniformly bounded if and only if GC /HC is spherical, or equivalently, HC acts on the full ﬂag variety GC /B with ﬁnitely manyorbits, where B is a Borel subgroup of GC . Thus, there exists a relationship between the uniform boundedness of the multiplicities in the regular representation
on G/H and the ﬁniteness of the number of HC -orbits on the full ﬂag variety
GC /B.
In this talk, we discuss a variant of this relationship for a generalized ﬂag
variety GC /QC , where Q is parabolic subgroup of G. More precisely, we provethat the multiplicities in C ∞ (G/H) of degenerate principal series representationsof G induced from Q are uniformly bounded if HC acts on GC /QC with ﬁnitely
many orbits using the theory of holonomic D-modules.