博士論文
論文題目
Visible actions of reductive algebraic groups on complex algebraic
varieties（簡約代数群の複素代数多様体への可視的作用について）
田中 雄一郎Dedicated to my grandfathers Kesao Tanaka and Morio Saeki, grandmother
Iku Saeki and friend Yohei Sato with wishes for good health of mygrandmother Setsuko Tanaka and greatuncle Shin-nosuke Saeki.

1Acknowledgment
The author would like to express his deepest appreciation to his research supervisor Professor Toshiyuki Kobayashi for his continuous and generous support, and patient and warm
encouragement. Associate Professor Hideko Sekiguchi showed to the author the ﬁrst stageof representation theory, and looked over him with warm hart. Therefore he would like toexpress his sincere gratitude to her. He is very grateful to Professor Toshitake Kohno forhis generous support and for giving him several opportunities to talk and to discuss withhis students. He is deeply grateful to Professor Toshihiko Matsuki for showing his interest
to this work and for his constant encouragement. He is very thankful to Professors Kyo
Nishiyama and Hiroyuki Ochiai for giving him insightful comments and suggestions. He
would like to thank Professors Susumu Ariki, Takashi Hashimoto, Yasufumi Hashimoto,
Masatoshi Iida, Katsuhiko Kikuchi, Kazufumi Kimoto, Takeyoshi Kogiso, Motoko Kotani,
Ivaı̈lo M. Mladenov, Kyoji Saito, Takashi Sakai, Reiho Sakamoto, Tatsuya Tate for giving him opportunities to talk about his research. He appreciates very much Associate
Professor Taro Yoshino, Tenured Lecturer Atsumu Sasaki and Assistant Professor Toshihisa Kubo for many invaluable discussions in Kobayashi lab seminar. He wants to thankmembers of the student room 406 of the Graduate School of Mathematical Science Building of The University of Tokyo for warm and generous atmosphere. Special thanks go
to Dr. Takayuki Okuda, Yoshiki Oshima and Yuya Matsumoto, Mr. Naoya Umezaki,
Yuki Fujii, Tatsuyuki Hikita, Takanori Masuda, Kotaro Mitome, Yusuke Nemoto, Fumihiko Sanda, Barmeier Severin, Yuya Takayama, Tomohiro Tsuda, Kohei Yahiro, Yuri
Yatagawa, Masatoshi Kitagawa, Ryosuke Nakahama, Koichi Tojo, Yosuke Morita, Oleksii
Leontiev, Naoya Shimamoto and Taito Tauchi for their support and encouragement.
The author is deeply grateful to people, especially Mr. Yukio Katayama and Akira
Suzuki, who clean up bath rooms and seminar rooms every early morning, and the aroundof the math. building every day. Thanks to them, bath rooms, seminar rooms and every
area of the building are always comfortable, and he can study with ease every day. Their(loud) “Good morning!” wake his brain up. He is also grateful to guards of the math.
building of The University of Tokyo, who pass the time of day in early morning. He is
thankful to Ms. Hiroko Furuno, Hanako Tagaya and Kanako Yamagishi for her support
in the work of teaching assistant. He is also grateful to Ms. Hiroe Idei for her support for
the use of his grant, and to Ms. Aki Koyama, Kanae Sato and Kaoru Takasugi for theirsupport in the course of the Program for Leading Graduate School. Thanks also go to Ms.
Akiko Nakamura for her help to his friends from abroad, and to Ms. Aki Nakagawa for her
generous support with network in the math. building. He wants to thank Ms. Nobue Fukui
for her support with Annual Report every year. Ms. Noriko Sato and Etsuko Hamada gave
him some snacks and coﬀees which help his brain much, so he is grateful to them. He is
thankful to Ms. Mari Ushijima for her support for his talk in the conference AAS2013. He
would like to express his sincere gratitude to all the staﬀs of the math. building of The
University of Tokyo for their generous, continuous and warm support. This work can not
be completed without their help. Also he is thankful to the workers in Mokku-Mokku, whooﬀer big lunch-boxes at a low price with an additional box ﬁlled with vegetables for free(maybe this is unusual and I should not spread this, although).
The author would like to express his sincere gratitude to his friends, especially to
Yasuaki Arima, Toshishi Matsuo, Syo Ohata, Tadaki Takahashi, Haruki Watanabe, Kosuke Yusa, Hiroshi Arakawa, Kei Katano, Yuta Meguro, Akihiro Miyata, Hiroaki Naga2

sawa, Keisuke Otake, Kenji Sasaki, Satoru Shimoyama, Hideyuki Sugita, Takashi Takeda,
Kazuhiro Tekuramori, Satoru Tomiya and the sixth members of the Taido club of The
Univ. Tokyo for their kindness and invaluable encouragement.
The author would like to express his greatest appreciation to his family, Saichiro,
Mariko, Tomomi and Risa for their continuous, invaluable and warmest support and encouragement.
This work was partly supported by the Grant-in-Aid for JSPS Fellows 24·6877, and theProgram for Leading Graduate Schools, MEXT, Japan.

3Chapter 1
Introduction
We study visible actions on complex algebraic varieties, and the main result is a classiﬁcation of visible actions on generalized ﬂag varieties.
Definition 1.0.1 (Kobayashi [Ko2]). We say a holomorphic action of a Lie group G on acomplex manifold X is strongly visible if the following two conditions are satisﬁed:
1. There exists a real submanifold S (called a “slice”) such thatX ′ := G · S is an open subset of X.
2. There exists an anti-holomorphic diﬀeomorphism σ of X ′ such thatσ|S = idS ,σ(G · x) = G · x for any x ∈ X ′ .
In the above setting, we say the action of G on X is S-visible. This terminology will
be used also if S is just a subset of X.
Definition 1.0.2 (Kobayashi [Ko2]). We say a holomorphic action of a Lie group G on acomplex manifold X is previsible if the condition (1) of Deﬁnition 1.0.1 is satisﬁed for a
totally real submanifold S of X.
The notion of visible actions on complex manifolds was introduced by T. Kobayashi[Ko2] with the aim of uniform treatment of multiplicity-free representations of Lie groups.
Definition 1.0.3. We say a unitary representation V of a locally compact group G ismultiplicity-free if the ring EndG (V ) of intertwining operators on V is commutative.
There are various kinds of multiplicity-free representations (c.f. [BR, HU, Ka, VK]), andfor the proof of the multiplicity-freeness property of representations, typical approaches are
the following: verifying the existence of an open orbit of a Borel subgroup; using a combinatorial method (computing or estimating coeﬃcients of the character of a representation).
These two approaches work very well for (the direct sum of) ﬁnite dimensional representations, but it would be hard to apply them to the inﬁnite dimensional representations
with continuous spectra. A new approach has been introduced by Kobayashi, namely, thepropagation theorem of the multiplicity-freeness property under visible actions:4Fact 1.0.4 (Kobayashi [Ko3]). Let G be a Lie group and W a G-equivariant Hermitian holomorphic vector bundle on a connected complex manifold X. Let V be a unitary
representation of G. If the following conditions from (0) to (3) are satisﬁed, then V ismultiplicity-free as a representation of G.
(0) There exists a continuous and injective G-intertwining operator from V to the spaceO(X, W) of holomorphic sections of W.
(1) The action of G on X is S-visible. That is, there exist a subset S ⊂ X and an antiholomorphic diﬀeomorphism σ of X ′ satisfying the conditions given in Deﬁnition
1.0.1. Further, there exists an automorphism σ̂ of G such that σ(g · x) = σ̂(g) · σ(x)for any g ∈ G and x ∈ X ′ .
(2) For any x ∈ S, the ﬁber Wx at x decomposes as the multiplicity-free ⊕sum of irreducible
unitary representations of the isotropy subgroup Gx . Let Wx =Wx(i) denote1≤i≤n(x)the irreducible decomposition of Wx .

(i)

(i)

(3) σ lifts to an anti-holomorphic automorphism σ̃ of W and satisﬁes σ̃(Wx ) = Wx
for each x ∈ S (1 ≤ i ≤ n(x)).

The advantage of this new approach is that not only ﬁnite dimensional cases but also
inﬁnite dimensional (both discrete and continuous spectra) cases can be applied by thismethod. Indeed, we can see in the statement of the above theorem that we do not need toassumeG is compact, reductive,
V is of ﬁnite-dimensional, discretely decomposable, orX is compact.
In the following, we quote a few examples of applications of Fact 1.0.4 from [Ko2]. The ﬁrst
example is an inﬁnite dimensional unitary representation with only continuous spectrum.
Example 1.0.5. Let G be a semisimple Lie group and K a maximal compact subgroupof G. Then it is well-known that the space L2 (G/K) of square integrable functions on the
Riemannian symmetric space G/K is multiplicity-free (see [Wo] for example). We can also
prove the multiplicity-freeness property by combining Fact 1.0.4 with the following facts.
• The G-action on the complexiﬁcation GC /KC is strongly visible by Kobayashi [Ko2].
• Let U be the complex crown of G/K, which was introduced by Akhiezer and Gindikin[AG]. Then there exists a G-embedding L2 (G/K) ,→ O(U ) by Krötz and Stanton[KS].
Next, we give an example of a multiplicity-free representation arising from a visible
action of a semisimple Lie group on a Hermitian symmetric space.
Example 1.0.6. Let G be a simple Lie group of Hermitian type, K a maximal compact
subgroup and H a symmetric subgroup of G, i.e., H is an open subgroup of the τ -ﬁxed
points subgroup Gτ for an involution τ of G. Let π be a unitary highest weight representation of the scalar type of G. Then the restriction of π to H is multiplicity-free [Ko5] byFact 1.0.4 combined with the following facts.
5• π can be realized in the space O(G/K, L) of holomorphic sections of a G-equivariant
holomorphic line bundle L on the Hermitian symmetric space G/K.
• The H-action on G/K is strongly visible by Kobayashi [Ko5] by the Cartan decomposition G = HAK in the symmetric setting (see Flensted-Jensen [Fl1], Hoogenboom[Ho] and Matsuki [Ma1, Ma2]).
As the last example, we show a multiplicity-free representation of a non-reductive Liegroup.
Example 1.0.7. Let G, K and π as in Example 1.0.6. Let N be a maximal unipotent
subgroup of G. Then the restriction of π to N is multiplicity-free by Fact 1.0.4 combinedwith the facts that π can be realized in O(G/K, L) for a G-equivariant holomorphic line
bundle L on G/K, and that the action of N on G/K is strongly visible by Kobayashi [Ko2]by the Iwasawa decomposition G = N AK.
As these examples show, we can obtain multiplicity-free representations from a visible
action of a Lie group. Therefore it would be natural to try to ﬁnd, or even classify, visible
actions. In the following, we exhibit preceding results on a classiﬁcation problem of visible
actions. We ﬁrstly state a result on visible actions on symmetric spaces.
Fact 1.0.8 (Kobayashi [Ko5]). Let (G, K) be a Hermitian symmetric pair and (G, H) asymmetric pair. Then H acts on the Hermitian symmetric space G/K strongly visibly.
The next result concerns the visibility of linear actions. Let GC be a connected complex
reductive algebraic group and V a ﬁnite-dimensional representation of GC .
Definition 1.0.9. We say V is a linear multiplicity-free space of GC if the space C[V ] ofpolynomials on V is multiplicity-free as a representation of GC .
Fact 1.0.10 (Sasaki [Sa1, Sa4]). Let V be a linear multiplicity-free space of GC . Then acompact real form U of GC acts on V strongly visibly.
Remark 1.0.11. We note that if U acts on a representation V of GC strongly visibly, thenV is a linear multiplicity-free space of GC by Fact 1.0.4.
A linear multiplicity-free space is a special case of smooth aﬃne spherical varieties.
Let GC be a connected complex reductive algebraic group and X a connected complex
algebraic GC -variety.
Definition 1.0.12. We say X is a spherical variety of GC if a Borel subgroup B of GC has
an open orbit on X.
A typical example of spherical varieties is a complex symmetric space (e.g. GC =GL(n, C) and X = GL(n, C)/(GL(m, C) × GL(n − m, C))). The third result deals withvisible actions on aﬃne homogeneous spherical varieties.
Fact 1.0.13 (Sasaki [Sa2, Sa3, Sa5]). Let GC /HC be one of the following aﬃne homogeneous spherical varieties:
SL(m + n, C)/(SL(m, C) × SL(n, C)) (m ̸= n),Spin(4n + 2, C)/SL(2n + 1, C),
SL(2n + 1, C)/Sp(n, C),
E6 (C)/Spin(10, C),SO(8, C)/G2 (C).
6

Then the action of a compact real form U of GC on GC /HC is strongly visible.
Lastly we state a classiﬁcation result on visible actions on generalized ﬂag varieties oftype A, which is the prototype of the main result of this paper. Let G = U(n) and L, H
Levi subgroups of G. Kobayashi [Ko4] classiﬁed the triple (G, H, L) such that the following
actions are strongly visible (we denote by ∆(G) the diagonal subgroup of G × G).
L ↷ G/H, H ↷ G/L, ∆(G) ↷ (G × G)/(H × L).
In fact, all the above three actions are strongly visible if and only if at least one of those isstrongly visible [Ko2]. The visibility of the three actions on generalized ﬂag varieties wasproved by giving a generalized Cartan decomposition:
Definition 1.0.14. Let G be a connected compact Lie group, T a maximal torus and H, L
Levi subgroups of G, which contain T . We take a Chevalley–Weyl involution σ of G withrespect to T . If the multiplication mapping
L×B×H →Gis surjective for a subset B of the σ-ﬁxed points subgroup Gσ , then we say the decomposition
G = LBH is a generalized Cartan decomposition.
Definition 1.0.15. An involution σ of a compact Lie group G is said to be a Chevalley–
Weyl involution if there exists a maximal torus T of G such that σ(t) = t−1 for any t ∈ T .
The deﬁnition of a generalized Cartan decomposition comes from that of a visible action.
Let us explain. We retain the setting of Deﬁnition 1.0.14. Suppose that G = LBH holds
for some B ⊂ Gσ . Since σ acts on generalized ﬂag varieties
G/H,(G × G)/(H × L)

G/L,

as anti-holomorphic diﬀeomorphisms, we can obtain three strongly visible actions.
L ↷ G/H,H ↷ G/L,∆(G) ↷ (G × G)/(H × L).

Furthermore, we can obtain three multiplicity-free theorems by using Fact 1.0.4.
indG
H χH |L ,indG
L χL |H ,G
indG
H χH ⊗ indL χL .

G
Here indG
H χH and indL χL denote the holomorphically induced representations from unitary characters χH and χL of H and L, respectively. As we saw, one generalized Cartan
decomposition leads us to three strongly visible actions, and three multiplicity-free theorems (Kobayashi’s triunity principle [Ko1]).
As the name indicates, the decomposition G = LBH can be regarded as a generalizationof the Cartan decomposition. Under the assumption that both (G, H) and (G, L) aresymmetric pairs, the decomposition theorem of the form G = LBH or its variants has
been well-established: G = KAK with K compact by É. Cartan, G = KAH with G, H
non-compact and K compact by Flensted-Jensen [Fl1], G = KAH with G compact by
Hoogenboom [Ho], and the double coset decomposition L\G/H by Matsuki [Ma1, Ma2].
We note that in our setting the subgroups L and H of G are not necessarily symmetric.

7

1.1

Main result 1: Classification of visible triples

The theorem below gives a classiﬁcation of generalized Cartan decompositions (Deﬁnition
1.0.14).
Theorem 1.1.1 ([Ta2, Ta3, Ta4, Ta5]). Let G be a connected compact simple Lie group, Ta maximal torus, Π a simple system and L1 , L2 Levi subgroups of G, whose simple systems
are given by proper subsets Π1 , Π2 of Π. Let σ be a Chevalley–Weyl involution of G withrespect to T . Then the triples (G, L1 , L2 ) listed below exhaust all the triples such that themultiplication mapping
L1 × B × L2 → G
is surjective for a subset B of Gσ .
Remark 1.1.2. For the type A simple Lie groups (or G = U(n)), this theorem was proved
by Kobayashi [Ko4].
In the following, we specify only the types of simple Lie groups G since our classiﬁcation
is independent of coverings, and list pairs (Π1 , Π2 ) of proper subsets of Π instead of pairs(L1 , L2 ) of Levi subgroups of G. Also, we put (Πj )c := Π \ Πj (j = 1, 2).
Classification for type An [Ko4]◦◦◦α1 α2 α3
Hermitian type:◦◦◦αn−2 αn−1 αnI. (Π1 )c = {αi }, (Π2 )c = {αj }.
Non-Hermitian type:I. (Π1 )c = {αi , αj }, (Π2 )c = {αk }, min{p, n + 1 − p} = 1 or i = j ± 1.
p=i,jII. (Π1 )c = {αi , αj }, (Π2 )c = {αk }, min{k, n + 1 − k} = 2.
III. (Π1 )c = {αl },Π2 : arbitrary, l = 1 or n.

Here i, j, k satisfy 1 ≤ i, j, k ≤ n.
Classification for type Bn◦◦◦
α1 α2 α3
Hermitian type:◦+3◦◦
αn−2 αn−1 αnI. (Π1 )c = {α1 }, (Π2 )c = {α1 }.
Non-Hermitian type:I. (Π1 )c = {αn }, (Π2 )c = {αn }.
II. (Π1 )c = {α1 }, (Π2 )c = {αi }, 2 ≤ i ≤ n.

8

Classification for type Cn◦◦ks◦αn−2 αn−1 αn◦◦◦α1 α2 α3
Hermitian type:I. (Π1 )c = {αn }, (Π2 )c = {αn }.
Non-Hermitian type:I. (Π1 )c = {α1 }, (Π2 )c = {αi }, 1 ≤ i ≤ n.
Classification for type Dn
◦
◦
α1 α2
Hermitian type:

◦ αn
~~~
~
@
◦
◦ @@
αn−3 αn−2 @◦ αn−1

I. (Π1 )c = {αi }, (Π2 )c = {αj }, i, j ∈ {1, n − 1, n}.
Non-Hermitian type:I. (Π1 )c = {α1 }, (Π2 )c = {αj }, j ̸= 1, n − 1, n.
II. (Π1 )c = {αi }, (Π2 )c = {αj }, i ∈ {n − 1, n}, j ∈ {2, 3}.
III. (Π1 )c = {αi }, (Π2 )c = {αj , αk }, i ∈ {n − 1, n}, j, k ∈ {1, n − 1, n}.
IV. (Π1 )c = {αi }, (Π2 )c = {α1 , α2 }, i ∈ {n − 1, n}.
V. (Π1 )c = {α1 }, (Π2 )c = {αj , αk }, j or k ∈ {n − 1, n}.
VI. (Π1 )c = {αi }, (Π2 )c = {α2 , αj }, n = 4, (i, j) = (3, 4) or (4, 3).
Classification for type E6
α2
◦
◦
◦
◦
α1 α3 α4
Hermitian type:

◦
α5

◦
α6

I. (Π1 )c = {αi }, (Π2 )c = {αj }, i, j ∈ {1, 6}.
Non-Hermitian type:I. (Π1 )c = {αi }, (Π2 )c = {α1 , α6 }, i = 1 or 6.
II. (Π1 )c = {αi }, (Π2 )c = {αj }, i = 1 or 6, j ̸= 1, 4, 6.

9

Classification for type E7
α2
◦
◦
◦
◦
α1 α3 α4
Hermitian type:

◦
α5

◦
α6

◦
α7

I. (Π1 )c = {α7 }, (Π2 )c = {α7 }.
Non-Hermitian type:I. (Π1 )c = {α7 }, (Π2 )c = {αi }, i = 1 or 2.
Classification for type E8 , F4 , G2
There is no pair (Π1 , Π2 ) of proper subsets of Π such that G = L1 Gσ L2 holds.
For the proof of suﬃciency of Theorem 1.1.1, we use the herringbone stitch method
introduced by Kobayashi [Ko4], which reduces unknown decompositions to the known
decomposition in the symmetric case. This method enables us to obtain a generalized
Cartan decomposition G = L1 BL2 with B ⊂ Gσ (Deﬁnition 1.0.14). For the proof ofnecessity in the classical case, we prove that G ̸= L1 Gσ L2 for any pair (Π1 , Π2 ) which is
not in the list in Theorem 1.1.1 by using invariant theory for quivers associated to Levi
subgroups. For the proof in the exceptional case, we use Fact 1.0.4 and Stembridge’sclassiﬁcation of multiplicity-free tensor product representations ([St2]).

1.2

Main result 2: Classification of visible actions on
generalized flag varieties

As we explained before, one generalized Cartan decomposition (Deﬁnition 1.0.14) leads us
to three strongly visible actions. The following corollary shows that the converse is also
true in our setting. Therefore we can obtain a classiﬁcation of visible actions on generalized
ﬂag varieties from Theorem 1.1.1.
Corollary 1.2.1 ([Ta1]). We retain the setting of Theorem 1.1.1. We denote by GC and(Lj )C the complexiﬁcations of G and Lj , respectively (j = 1, 2). We let Pj be a parabolic
subgroup of GC with Levi subgroup (Lj )C , and put Pj = GC /Pj (j = 1, 2). Then thefollowing eleven conditions are equivalent.
(i) The multiplication mapping L1 × Gσ × L2 → G is surjective.
(ii) The natural action L1 ↷ P2 is strongly visible.
(iii) The natural action L2 ↷ P1 is strongly visible.
(iv) The diagonal action ∆(G) ↷ P1 × P2 is strongly visible.
(v) Any irreducible representation of G, which belongs to P2 -series is multiplicity-freewhen restricted to L1 .
10

(vi) Any irreducible representation of G, which belongs to P1 -series is multiplicity-freewhen restricted to L2 .
(vii) The tensor product of arbitrary two irreducible representations π1 and π2 of G, which
belong to P1 and P2 -series, respectively, is multiplicity-free.
(viii) P2 is a spherical variety of (L1 )C .
(ix) P1 is a spherical variety of (L2 )C .
(x) P1 × P2 is a spherical variety of ∆(GC ).
(xi) The pair (Π1 , Π2 ) is one of the entries listed in Theorem 1.1.1 up to switch of thefactors.
Here an irreducible representation of G is in Pj -series if it is a holomorphically induced
representation from a unitary character of the Levi subgroup Lj (j = 1, 2).
Proof. ∗ We prove that Theorem 1.1.1 implies this corollary. The strategy of the proof issummarized in the below diagram.
⇔(vii) · · · · · · multiplicity-free⇔(xi) · · · · · · classiﬁcation of (L1 , L2 )

⇐
⇔

⇐
⇐
⇔

(iii) · · · · · · visible action

⇐

(iv)

⇔

(ii)

⇒

⇐

(i) · · · · · · Cartan decomposition(viii)(x)(v) ⇔ (vii) ⇔ (vi)· · · · · · multiplicity-free(ix)· · · · · · spherical action.

The implication (vii) ⇒ (xi) can be veriﬁed by comparing Stembridge’s classiﬁcation[St2] with Theorem 1.1.1. The converse implication (xi) ⇒ (vii) follows from Fact 1.0.4.
The equivalence (xi) ⇔ (i) is Theorem 1.1.1. The implications (i) ⇒ (ii), (i) ⇒ (iii) and(i) ⇒ (iv) are the triunity of visibility ([Ko1]). Each of the three implications (ii) ⇒ (v),(iii) ⇒ (vi) and (iv) ⇒ (vii) is followed by Fact 1.0.4. As in the proof of [Ko2, Corollary15], we see that a result of Vinberg and Kimel’fel’d [VK, Corollary 1] implies the threeequivalences (v) ⇔ (viii), (vi) ⇔ (ix) and (vii) ⇔ (x). The equivalence (v) ⇔ (vii) ⇔ (vi)on the multiplicity-freeness property of representations follows from a result of Stembridge[St2, Corollary 2.5]. This completes the proof of the corollary.
Remark 1.2.2. For the type A simple Lie groups (or G = U(n)), this corollary was proved
by Kobayashi [Ko2].
Remark 1.2.3. Littelmann [Li] classiﬁed for any simple algebraic group G over any algebraically closed ﬁeld of characteristic zero, all the pairs of maximal parabolic subgroupsPω and Pω′ corresponding to fundamental weights ω and ω ′ , respectively, such that the∗This proof for Corollary 1.2.1 is quoted from [Ta1].

11

tensor product representation Vnω ⊗ Vmω′ decomposes multiplicity-freely for arbitrary nonnegative integers n and m. Moreover, he found the branching rules of Vnω ⊗ Vmω′ and therestriction of Vnω to the maximal Levi subgroup Lω′ of Pω′ for any pair (ω, ω ′ ) that admitsa ∆(G)-spherical action on G/Pω × G/Pω′ .
Remark 1.2.4. Stembridge [St2] gave a complete list of a pair (µ, ν) of highest weights
such that the corresponding tensor product representation Vµ ⊗ Vν is multiplicity-free forany complex simple Lie algebra g. His method was combinatorial and not based on the
Borel–Weil realization of irreducible representations. He also classiﬁed all the pairs (µ, l)of highest weights µ and Levi subalgebras l of g with the restrictions Vµ |l multiplicity-free.

1.3

Main result 3: Seeds and visible actions for theorthogonal group

As we mentioned in Remark 1.2.4, ﬁnite dimensional multiplicity-free tensor product representations were classiﬁed by Stembridge [St2]. By using the notion of visible actions
on complex manifolds, we would be able to, and indeed can in the types A, B and D
cases, understand his classiﬁcation more deeply. By Fact 1.0.4, we can reduce complicatedmultiplicity-free theorems to a pair of data:
visible actions on complex manifolds, andmuch simpler multiplicity-free representations (seeds of multiplicity-free representations).
For the type A simple Lie groups, Kobayashi found the following seeds of multiplicity-free
representations that combined with visible actions can produce all the cases of the pair of
two representations (V1 , V2 ) of U(n) such that V1 ⊗ V2 is multiplicity-free [Ko1].
• One-dimensional representations.
• (U(n) ↓ Tn ) The restriction of an alternating tensor product representation Λk (Cn ).
• (U(n) ↓ Tn ) The restriction of a symmetric tensor product representation S k (Cn ).
• (U(n) ↓ U(n1 ) × U(n2 ) × U(n3 )) The restriction of an irreducible representation V2ωk(n = n1 + n2 + n3 ).
Here Vλ denotes an irreducible representation of U(n) with highest weight λ and {ωk }1≤k≤n−1
is the set of fundamental weights of U(n). On the other hand, he classiﬁed in [Ko4] visible
actions on generalized ﬂag varieties of type A as listed in Theorem 1.1.1. By combining
the above seeds of multiplicity-free representations with the visible actions and using histriunity principle, Kobayashi constructed all the multiplicity-free tensor product representations of U(n) [Ko1]. In this paper we construct all the multiplicity-free tensor productrepresentations for SO(N ) and its covering group Spin(N ) by following Kobayashi’s argument for U(n). In our case, visible actions come from triples (G, L1 , L2 ) for G = Spin(N )listed in Theorem 1.1.1 as in the case of the type A groups. On the other hand, seedsof multiplicity-free tensor product representations arise only from one-dimensional representations, alternating tensor product representations and spin representations. These are
12exhibited in Proposition 1.3.1. We can see how to combine those visible actions and seeds
to obtain multiplicity-free tensor product representations in Theorem 1.3.2.
We denote by Π = {αi }1≤i≤[N/2] (see Theorem 1.1.1 for the labeling of the Dynkin
diagrams) a simple system of the root system of G = Spin(N ) with respect to its maximal
torus T , and by {Hi }1≤i≤[N/2] the dual basis of Π. We deﬁne a subgroup M of Spin(2n + 1)
as follows.
{(√)}
M := exp −1mπH1 1≤m≤4 · Spin(2n − 1),(1.3.1)
where exp denotes the exponential mapping, and the simple system of Spin(2n−1) is givenby {αk ∈ Π : 2 ≤ k ≤ n}.
Proposition 1.3.1. We denote by 1, CN and SpinN for the one-dimensional trivial representation, the natural representation and the spin representation of Spin(N ), respectively.
Then the following hold.
(1) One-dimensional representations are multiplicity-free.
(2) 1, CN and SpinN are multiplicity-free as representations of a maximal torus T ofSpin(N ).
(3) Λi (CN ) is multiplicity-free as a representation of a maximal Levi subgroup U(j) ×SO(N − 2j) of SO(N ) (when N is even and i = N/2, we replace ΛN/2 (CN ) by
its SO(N )-irreducible constituent whose highest weight is 2ωN/2−1 or 2ωN/2 ) if thefollowing condition (3-1) or (3-2) is satisﬁed (1 ≤ i, j ≤ [N/2]).
(3-1) N is odd.
(3-2) N is even, and i, j satisfy i + j ≤ N/2, j = N/2 or i = N/2.
(4) SpinN is multiplicity-free as a representation of M , where N is odd and M as in(1.3.1).
The theorem below gives a geometric construction of all the multiplicity-free tensor
product representations for the orthogonal group. For a realization of irreducible representations of a compact Lie group, we use the Borel–Weil theory. Namely, we realize anirreducible representation of a compact Lie group G as the space O(G/L, W) of holomorphic sections of a vector bundle W on a generalized ﬂag variety G/L, which is associatedto an irreducible representation W of a Levi subgroup L of G.
Theorem 1.3.2. We let G = Spin(N ). For any two irreducible representations Vλ1 andVλ2 of G such that Vλ1 ⊗ Vλ2 is multiplicity-free, there exists a pair of• a generalized ﬂag variety (G × G)/(L1 × L2 ) with a strongly visible ∆(G)-action, and• irreducible representations (a seed given in Proposition 1.3.1) W1 and W2 of L1 andL2 , respectively,
such that Vλk ≃ O(G/Lk , Wk ) as G-modules (k = 1, 2).
The correspondence between the data (Lk , Wk ) of visible actions and seeds, and thehighest weights λk of Vλk (k = 1, 2) is given as in Tables 1.3.1–1.3.4 below. In the tables,
Cλ denotes a one-dimensional representation with weight λ, T a maximal torus of G andLλ a Levi subgroup of G, whose simple system is given by {αl ∈ Π : ⟨λ, α̌l ⟩ = 0} where α̌lis the coroot of αl (1 ≤ l ≤ [N/2]).
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L1
L λ1

L2
L λ2

Table 1.3.1: Line bundle type
W2 N
λ1
λ2
Cλ2 2n + 1 sω1
tωj
sωn
tωn
2n
sω1
tωj + uωn−δsωn−δ tω3 , tω1 + uω2 , tω1 + uωn−δ′or tωn−1 + uωn8sω5−ϵ tω2 + uω2+ϵW1
Cλ1

1 ≤ j ≤ n, s, t, u ∈ N, δ = 0 or 1, δ ′ = 0 or 1 and ϵ = 1 or 2.

L1
G

L1
G

Table 1.3.2: Weight multiplicity-free type
L2 W1 W2 N
λ1
λ2
T Vλ1 Cλ2 2n + 1 0, ω1 or ωnarbitrary2n0, ω1 , ωn−1 or ωn arbitrary
Table 1.3.3: Alternating tensor product type
L2 W1 W2 N
λ1
λ2
Condition
Lλ2 Vλ1 Cλ2 2n + 1 ωi or 2ωn tωj
2n
ωi
tωj
i+j ≤n
ωi
tωn−δ
2ωn−δ
tωj
1 ≤ i, j ≤ n, t ∈ N and δ = 0 or 1.

L1
L λ1

L2
Lωj

W1
Cλ1

Table 1.3.4: Spin typeW2
N
λ1
C(1/2+t)ωj ⊠ SpinN −2j 2n + 1 sω1λ2
ωn + tωj

1 ≤ j ≤ n − 1 and s, t ∈ N.

By virtue of Fact 1.0.4 and the triunity principle [Ko1], we obtain the following corollary.
This corollary was proved by Stembridge [St2] by a combinatorial method.
Corollary 1.3.3. We retain the notation of Theorem 1.3.2. For the data (L1 , L2 , N ,λ1 , λ2 ) of each row in Tables 1.3.1–1.3.4, the representations Vλ1 and Vλ2 of G decompose
multiplicity-freely when restricted to the subgroups L2 and L1 of G, respectively.
So far we have considered visible actions of Levi subgroups on generalized ﬂag varieties.
For a general spherical variety, we have the following result on the visibility of actions ofcompact Lie groups. Let U be a compact real form of a connected complex reductive
algebraic group GC , and X a GC -spherical variety. We denote by θ the Cartan involutionof GC , which corresponds to U , and by ν a Chevalley–Weyl involution of GC (i.e., ν is an
involution of GC , which satisﬁes ν(t) = t−1 for any element t ∈ TC for some maximal torus
TC ), which preserves U . We put ι = θ ◦ ν.

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Theorem 1.3.4. Assume that there exists a real structure µ on a GC -spherical variety X
compatible with ι and that the µ-ﬁxed points subset X µ is non-empty. Then a compact real
form U acts on X strongly visibly.
Here by a real structure on a complex manifold Z we mean an anti-holomorphic involution η : Z → Z [Ak, AC]. Also for a real structure η on a complex manifold Z with anaction of a group K, we say η is compatible with an automorphism ϕ of K if η satisﬁes
η(kz) = ϕ(k)η(z) for any k ∈ K and z ∈ Z. Combining Theorem 1.3.4 with Akhiezer’s
result [Ak] on the existence of compatible real structures on Stein manifolds, we obtain
Corollary 1.3.5. Let (GC , V ) be a linear multiplicity-free space. Then a compact real form
U acts on V strongly visibly.
Corollary 1.3.6. Let X be a smooth aﬃne GC -spherical variety. Then a compact real
form U acts on X strongly visibly.
Here a typical example of smooth aﬃne spherical varieties is a complex symmetricspace. On the other hand, we have the principal aﬃne space GC /N (N is a maximal
unipotent subgroup) as an example of non-aﬃne smooth spherical varieties. We remarkthat Corollary 1.3.5 was earlier proved by Sasaki (Fact 1.0.10) by constructing slices explicitly. By combining Theorem 1.3.4 with Akhiezer and Cupit-Foutou’s result [AC], we
also have
Corollary 1.3.7. Let X be a GC -wonderful variety. Then a compact real form U acts on
X strongly visibly.
Definition 1.3.8. A GC -variety X is said to be wonderful if• X is smooth and projective,• GC has an open orbit on X, whose complement is a union of ﬁnitely many smooth
prime divisors Xi (i ∈ I) with normal crossings, and• the closure of any GC -orbit on X is given as a partial intersection of Xi (i ∈ I).
To prove the visibility of actions of non-compact reductive groups on complex manifolds,
we use the following extension of a result of Matsuki [Ma1, Ma2]. Let L and H be reductive
subgroups of a connected real semisimple algebraic group G such that both GC /LC andGC /HC are GC -spherical varieties.
Theorem 1.3.9. There exist∪mﬁnitely many abelian subspaces ji of g and elements xi ofG (i = 1, . . . , m) such that i=1 LCi H contains an open dense subset of G, where Ci =exp(ji )xi .
We use this decomposition to show the previsibility of actions of non-compact reductive
groups.
Theorem 1.3.10. Let X be a GC -spherical variety and G a real form of inner type of GC .
Then G acts on X previsibly.
Here a real reductive Lie group is said to be of inner type if its Lie algebra has a compact
Cartan subalgebra.
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