The classification of thick representations of simple Lie groups
aa r X i v : . [ m a t h . R T ] A ug THE CLASSIFICATION OF THICK REPRESENTATIONS OFSIMPLE LIE GROUPS
KAZUNORI NAKAMOTO AND YASUHIRO OMODA
Abstract.
We characterize finite-dimensional thick representations over C ofconnected complex semi-simple Lie groups by irreducible representations which areweight multiplicity-free and whose weight posets are totally ordered sets. More-over, using this characterization, we give the classification of thick representationsover C of connected complex simple Lie groups. Introduction
In our previous paper [7], we have introduced m -thickness and thickness of grouprepresentations. Let ρ : G → GL( V ) be a finite-dimensional representation of agroup G . If for any subspaces V and V of V with dim V = m and dim V = dim V − m there exists g ∈ G such that ( ρ ( g ) V ) ⊕ V = V , we say that a representation ρ : G → GL( V ) is m -thick . We also say that a representation ρ : G → GL( V )is thick if ρ is m -thick for each 0 < m < dim V (Definition 2.1). Remark that1-thickness is equivalent to irreducibility (Proposition 2.8). Hence m -thickness is anatural generalization of irreducibility of group representations.Let G be a connected semi-simple Lie group over C , B a Borel subgroup of G , T a maximal torus which is contained in B . Denote their Lie algebras by g , b and t ,respectively. Let ρ : G → GL( V ) be a finite-dimensional irreducible representationof G over C . We denote the set of t -weights in V by W ( V ). Choosing a set of simpleroots for ( g , t ), we can regard W ( V ) as a partially ordered set (poset) with respectto the usual root order. We call it the weight poset . We say that a representation ρ : G → GL( V ) is weight multiplicity-free if the weight spaces in V are all one-dimensional. We give the following characterization of thickness. Theorem 1.1 (Theorem 3.5) . An irreducible representation ρ : G → GL( V ) of aconnected semi-simple Lie group G is thick if and only if it is weight multiplicity-freeand its weight poset is a totally ordered set. Using this characterization, we can classify the complex thick representations ofconnected semi-simple Lie groups.
Mathematics Subject Classification.
Primary 22E46; Secondary 22E47, 17B10.
Key words and phrases. thick representation, dense representation, simple Lie group.The first author was partially supported by JSPS KAKENHI Grant Number JP23540044,JP15K04814, JP20K03509.
Theorem 1.2 (Theorems 3.11 and 3.12) . If a representation of a connected semi-simple Lie group is thick, then it is geometrically equivalent to one of the followinglist: e, SL n ( n ≥ , S m SL ( m ≥ , SO n +1 ( n ≥ , Sp n ( n ≥ , G . Here the irreducible representation of a connected simple Lie group G of thehighest weight ω , where ω is the first fundamental weight, is denoted by G . Sim-ilarly, S m G stands for the m -th symmetric power of G . Let e denote the triv-ial 1-dimensional representation for any group G . For the definition of geometricequivalence, see Definition 3.8.We denote by ω i the i -th fundamental weight for a connected simple Lie group G . In §
3, all Lie groups are assumed to be over C and all representations arefinite-dimensional over C . 2. preliminaries A representation of a group G on a vector space V is a homomorphism ρ : G → GL( V ). Then such a map ρ gives V the structure of a G -module. We sometimes call V itself a representation of G and write gv for ρ ( g )( v ). We recall several definitionsand results in our previous paper [7]. Definition 2.1 ([7, Definition 2.1]) . Let G be a group. Let V be a finite-dimensionalvector space over a field k . We say that a representation ρ : G → GL( V ) is m -thick if for any subspaces V and V of V with dim V = m and dim V = dim V − m ,there exists g ∈ G such that ( ρ ( g ) V ) ⊕ V = V . We also say that a representation ρ : G → GL( V ) is thick if ρ is m -thick for each 0 < m < dim V . Definition 2.2 ([7, Definition 2.3]) . Let G be a group. Let V be a finite-dimensionalvector space over a field k . We say that a representation ρ : G → GL( V ) is m -dense if the induced representation ∧ m ρ : G → GL( ∧ m V ) is irreducible. We also say thata representation ρ : G → GL( V ) is dense if ρ is m -dense for each 0 < m < dim V .We show several examples. See [7] for details. Example 2.3 ( cf. [7, Proposition 6.5]) . Let V be the standard representation ofSL n and V ∗ the dual representation of V . Then V and V ∗ are dense. Example 2.4 ([7, Proposition 6.10]) . The standard representation of SO n is m -dense for each 0 < m < n with m = n , but not n -thick. Example 2.5 ([7, Proposition 6.11]) . The standard representation of SO n +1 isdense. Example 2.6 ([7, Proposition 6.18]) . The standard representation of Sp n is thick,but not m -dense for each 1 < m < n − HE CLASSIFICATION OF THICK REPRESENTATIONS OF SIMPLE LIE GROUPS 3
Let V be a finite-dimensional representation of a group G . For positive integers i and j with i + j = dim V , let us consider the G -equivariant perfect pairing ∧ i V ⊗∧ j V ∧ −→ ∧ dim V V ∼ = k . For a G -invariant subspace W of ∧ i V , put W ⊥ := { y ∈∧ j V | x ∧ y = 0 for any x ∈ W } . Then W ⊥ is also G -invariant. In particular, ∧ i V is irreducible if and only if so is ∧ j V . Proposition 2.7 ([7, Proposition 2.6]) . Let V be an n -dimensional representationof a group G . For each < m < n , V is m -thick ( resp. m -dense ) if and only if V is ( n − m ) -thick ( resp. ( n − m ) -dense ) . Proposition 2.8 ([7, Proposition 2.7]) . For any finite-dimensional representation V of a group G , the following implications hold for < m < dim V : m -dense = ⇒ m -thick ⇓ -dense ⇐⇒ -thick ⇐⇒ irreducible . Corollary 2.9 ([7, Corollary 2.8]) . For any finite-dimensional representation of agroup G , the following implications hold : dense ⇒ thick ⇒ irreducible. Corollary 2.10 ([7, Corollary 2.9]) . For any representation V of a group G with dim V ≤ , the following implications hold : dense ⇔ thick ⇔ irreducible. Definition 2.11 ([7, Definition 2.10]) . Let V be an n -dimensional vector space overa field k . For a d -dimensional subspace V ′ of V with 0 < d < n , we can considera point [ ∧ d V ′ ] in the projective space P ( ∧ d V ). In the sequel, we identify [ ∧ d V ′ ]with a non-zero vector ∧ d V ′ ∈ ∧ d V (which is determined by [ ∧ d V ′ ] up to scalar)for simplicity. For a vector subspace W ⊂ ∧ d V , we say that W is realizable if W contains a non-zero vector ∧ d V ′ obtained by a d -dimensional subspace V ′ of V .We have the following criterion of thickness. Proposition 2.12 ([7, Proposition 2.11]) . Let V be an n -dimensional representationof a group G . For < m < n , V is not m -thick if and only if there exist G -invariantrealizable subspaces W ⊆ ∧ m V and W ⊆ ∧ n − m V such that W ⊥ = W . The classification of thick representations of simple Lie groups
Let G be a connected semi-simple Lie group over the complex number field C , B a Borel subgroup of G , T a maximal torus which is contained in B , B − a Borelsubgroup of G opposite to B relative to T = B ∩ B − . Denote their Lie algebras by g , b , t and b − , respectively. Let V be a finite-dimensional irreducible representationof G over C . We will denote the set of t -weights in V by W ( V ). For any weight ϕ ∈ W ( V ), let V ϕ be the ϕ -weight space in V . Let Π be the set of simple roots KAZUNORI NAKAMOTO AND YASUHIRO OMODA and ∆ + the set of positive roots for ( g , b ). We can regard W ( V ) as a partiallyordered set (poset) with respect to the usual root order. More precisely, µ > γ ifand only if µ − γ is a nonzero sum of simple roots with nonnegative coefficients.In particular, if µ − γ is a simple root, we say that µ covers γ . We call W ( V ) the weight poset . We say that a representation V of G is weight multiplicity-free ( WMF )if the weight spaces in V are all one-dimensional. Howe [3] classified the irreduciblerepresentations of connected simple Lie groups which are weight multiplicity-free. Proposition 3.1.
If a representation V of G is thick, it is weight multiplicity-free.Proof. Assume that V is not WMF. Then there exists a weight ϕ ∈ W ( V ) such thatthe dimension of V ϕ is larger than one. Let W + ( ϕ ) be the set of all weights strictlylarger than ϕ , and Y + ( ϕ ) the subspace which is spanned by all weight spaces forweights in W + ( ϕ ). Because the dimension of V ϕ is larger than one, we can choose twolinear independent ϕ -weight vectors v and w . Let W a,bϕ (+) be C ( av + bw ) ⊕ Y + ( ϕ )for a, b ∈ C . The subspace W a,bϕ (+) is B -invariant. Let n be the dimension of V ,and d the dimension of W a,bϕ (+) for ( a, b ) ∈ C \ { (0 , } . The elements ∧ d W a,bϕ (+)for ( a, b ) ∈ C \ { (0 , } are distinct B -eigenvectors in ∧ d V with the same weight.Let U a,bϕ (+) be the irreducible G -submodule in ∧ d V with the highest weight vector ∧ d W a,bϕ (+). Let U ϕ (+) be the direct sum U , ϕ (+) ⊕ U , ϕ (+) ⊂ ∧ d V . Any irreducible G -submodule of U ϕ (+) is equal to U a,bϕ (+) for some ( a, b ) ∈ C \ { (0 , } . Hence anyirreducible G -submodule of U ϕ (+) is realizable.Let Y − ( ϕ ) be the subspace which is spanned by all weight spaces for weightsin W ( V ) \ { W + ( ϕ ) , ϕ } . We take a basis { v, w, u , . . . , u s } for V ϕ which contains v, w . Let W ϕ ( − ) be the subspace which is spanned by { w, u , . . . , u s } and Y − ( ϕ ).The subspace W ϕ ( − ) is invariant under the action of the opposite Borel subgroup B − . The equalities dim W ϕ ( − ) = dim V − dim W a,bϕ (+) = n − d hold for ( a, b ) ∈ C \ { (0 , } . Then ∧ n − d W ϕ ( − ) is a B − -eigenvector in ∧ n − d V . Let U ϕ ( − ) be theirreducible G -submodule with the lowest weight vector ∧ n − d W ϕ ( − ) in ∧ n − d V . Ob-viously, U ϕ ( − ) is realizable. The irreducibility of U ϕ ( − ) shows the irreducibility of ∧ d V / ( U ϕ ( − )) ⊥ . Then ( U ϕ ( − )) ⊥ ∩ U ϕ (+) = { } because U ϕ (+) is not irreducible.Hence ( U ϕ ( − )) ⊥ ∩ U ϕ (+) contains some realizable G -submodule U a,bϕ (+). Therefore( U ϕ ( − )) ⊥ is realizable. Putting W = ( U ϕ ( − )) ⊥ and W = U ϕ ( − ), we see that V isnot thick by Proposition 2.12. Hence if V is thick, then it is WMF. (cid:3) Proposition 3.2.
If a representation V of G is thick, its weight poset W ( V ) is atotally ordered set.Proof. Let V be a thick representation of G . By Proposition 3.1, V is WMF. Forany weight φ ∈ W ( V ), let W + ( φ ) be the set of all weights strictly larger than φ , and Y + ( φ ) the subspace which is spanned by all weight spaces for weights in W + ( φ ).Note that the irreducible representation V has a highest weight ω and that eachweight of V has the form ω − P li =1 m i α i ( m i ∈ N ), where Π = { α , . . . , α l } . HE CLASSIFICATION OF THICK REPRESENTATIONS OF SIMPLE LIE GROUPS 5
Suppose that the weight poset W ( V ) is not a totally ordered set. There exists apositive integer i such that W ( V ) has the i -th highest weight, but not the ( i + 1)-thhighest weight. Let ϕ be the i -th highest weight, and ψ , ψ maximal weights in W ( V ) \ ( W + ( ϕ ) ∪ { ϕ } ). Then the subset W + ( ϕ ) ∪ { ϕ } is a totally ordered set, ϕ covers ψ , ψ , and W + ( ψ ) = W + ( ψ ) = W + ( ϕ ) ∪ { ϕ } . Because V is WMF, thereexists a unique ψ i -weight vector v i up to scalar for each i = 1 ,
2. Let W ψ i (+) be C · v i ⊕ Y + ( ψ i ). The subspaces W ψ i (+) are B -invariant for each i = 1 ,
2. Let n be the dimension of V , and d the dimension of W ψ i (+). The elements ∧ d W ψ (+)and ∧ d W ψ (+) are distinct B -eigenvectors with distinct weights in ∧ d V . Let U ψ i (+)be the irreducible G -submodule of ∧ d V with the highest weight vector ∧ d W ψ i (+)for each i = 1 ,
2. Then U ψ (+) and U ψ (+) are realizable and not isomorphicto each other as G -modules. Let Y − ( ψ ) be the subspace which is spanned byall weight spaces for weights in W ( V ) \ ( W + ( ψ ) ∪ { ψ } ). The subspace Y − ( ψ )is invariant under the action of the opposite Borel subgroup B − . The equalitiesdim Y − ( ψ ) = dim V − dim W ψ (+) = n − d hold. Then ∧ n − d Y − ( ψ ) is a B − -eigenvector in ∧ n − d V . Let U ψ ( − ) be the irreducible G -submodule of ∧ n − d V withthe lowest weight vector ∧ n − d Y − ( ψ ). Then U ψ ( − ) is realizable. The irreducibilityof U ψ ( − ) shows the irreducibility of ∧ d V / ( U ψ ( − )) ⊥ . Then ( U ψ ( − )) ⊥ ∩ ( U ψ (+) ⊕ U ψ (+)) = { } . Because U ψ (+) is not isomorphic to U ψ (+), U ψ (+) ⊂ ( U ψ ( − )) ⊥ or U ψ (+) ⊂ ( U ψ ( − )) ⊥ . In particular, ( U ψ ( − )) ⊥ is realizable. Putting W =( U ψ ( − )) ⊥ and W = U ψ ( − ), we see that V is not thick by Proposition 2.12. Thisis a contradiction. Hence W ( V ) is a totally ordered set. (cid:3) Let us denote the Grassmann variety which is the set of all k -dimensional sub-spaces of a vector space V by Grass( k, V )( ⊂ P ( ∧ k V )). Lemma 3.3.
Let V be a representation of G , and W a G -invariant realizable sub-space of ∧ k V . Then there exists [ v ] ∈ P ( W ) ∩ Grass( k, V ) such that [ v ] is B -invariant.Proof. Let X be P ( W ) ∩ Grass( k, V ). Because W is realizable, X is not empty. Notethat X is G-invariant and compact. We take a G -orbit O in X whose dimension isminimal. The orbit O is closed and then compact. There is a parabolic subgroup P of G such that the orbit O is isomorphic to G/P . Then there is a point [ v ] ∈ O ⊂ P ( W ) ∩ Grass( k, V ) such that [ v ] is B -invariant. (cid:3) Lemma 3.4.
Assume that an irreducible representation V of G is weight multiplicity-free, its weight poset W ( V ) is a totally ordered set { ϕ > ϕ > · · · > ϕ n } , and W is a G -invariant realizable subspace of ∧ k V . Let v i be a nonzero vector in the ϕ i -weight space V ϕ i ( i = 1 , , . . . , n ) . Then W contains v ∧ v ∧ · · · ∧ v k and v n − ( k − ∧ v n − ( k − ∧ · · · ∧ v n .Proof. Because V is weight multiplicity-free, { v , . . . , v n } is a basis of V . By Lemma3.3, there exists [ v ] ∈ P ( W ) ∩ Grass( k, V ) such that v is a highest weight vector of KAZUNORI NAKAMOTO AND YASUHIRO OMODA an irreducible subrepresentation of W with respect to B . We can put v = ( p , v + p , v + · · · + p ,n v n ) ∧ ( p , v + p , v + · · · + p ,n v n )... ∧ ( p k, v + p k, v + · · · + p k,n v n )up to scalar multiplication, where P = ( p i,j ) is in reduced row echelon form. Remarkthat P is uniquely determined. Let X α be a root vector for a positive root α ∈ ∆ + .Then X α v = 0 holds for any α ∈ ∆ + . If p , = p , = · · · = p ,i = 0 and p ,i +1 = 1 for i ≥
1, there is a positive root α ∈ ∆ + such that X α v i +1 is cv i for a nonzero constant c . Then X α v is not 0. This is a contradiction. So p , = 1. Similarly, we can showthat p = · · · = p kk = 1. Because v is a highest weight vector, for any t ∈ t there is aconstant c such that tv = cv . Then by the uniqueness of P we can show that p ij = 0for i = 1 , . . . , k and j = k + 1 , . . . , n . Then v = v ∧ v ∧ · · · ∧ v k ∈ W . A similarargument with respect to B − shows that v n − ( k − ∧ v n − ( k − ∧ · · · ∧ v n ∈ W . (cid:3) Theorem 3.5.
An irreducible representation V of a connected semi-simple Lie group G is thick if and only if it is weight multiplicity-free and its weight poset is a totallyordered set.Proof. The “only if” part can be proved by Propositions 3.1 and 3.2. Let us provethe “if” part. Let us use the notations in Lemma 3.4. Assume that W ⊆ ∧ k V and W ⊆ ∧ n − k V are G -invariant realizable subspaces. By Lemma 3.4, v ∧ v ∧ · · ·∧ v k ∈ W and v k +1 ∧ v k +2 ∧· · ·∧ v n ∈ W . Since ( v ∧ v ∧· · ·∧ v k ) ∧ ( v k +1 ∧ v k +2 ∧· · ·∧ v n ) = 0, W ⊥ = W . By Proposition 2.12, V is thick. (cid:3) By [3, Theorem 4.6.3], we have Howe’s classification of irreducible representationsof connected simple Lie groups which are weight multiplicity-free. We also referto Panyushev’s paper [8, Table 1] for the weight posets of weight multiplicity-freerepresentations. Thus, we have
Theorem 3.6.
The thick representations of connected simple Lie groups are thoseon the following list: (1) the trivial -dimensional representation for any groups (2) A n ( n ≥ • the standard representation V for A n ( n ≥ with highest weight ω • the dual representation V ∗ of V for A n ( n ≥ with highest weight ω n • the symmetric tensor S m ( V ) ( m ≥ of V for A with highest weight mω (3) B n ( n ≥ • the standard representation V for B n ( n ≥ with highest weight ω • the spin representation for B with highest weight ω (4) C n ( n ≥ HE CLASSIFICATION OF THICK REPRESENTATIONS OF SIMPLE LIE GROUPS 7 • the standard representation V for C n ( n ≥ with highest weight ω (5) G • the -dimensional representation V for G with highest weight ω .Proof. By Theorem 3.5, it suffices to list up all irreducible representations which areweight multiplicity-free and whose weight posets are totally ordered sets. Using [3,Theorem 4.6.3] and [8, Table 1], we can obtain the list of thick representations ofconnected simple Lie groups. (cid:3)
We also have the list of dense representations:
Theorem 3.7.
The dense representations of connected simple Lie groups are thoseon the following list: (1) the trivial -dimensional representation for any groups (2) A n ( n ≥ • the standard representation V for A n ( n ≥ with highest weight ω • the dual representation V ∗ of V for A n ( n ≥ with highest weight ω n • the symmetric tensor S ( V ) of V for A with highest weight ω (3) B n ( n ≥ • the standard representation V for B n ( n ≥ with highest weight ω .Proof. It suffices to verify whether thick representations in the list of Theorems 3.6are dense or not. It is well-known that the standard representations V of A n and B n are dense. We also see that the dual representation V ∗ of V for A n is dense. (For A n , see Example 2.3 or [2, § B n , see Example 2.5 or [2, Theorem 19.14].)By Corollary 2.10, S ( V ) for A is dense since dim S ( V ) = 3.Conversely, let us show that S m ( V ) for A is not dense if m ≥
3. Let { ϕ > ϕ } be the weight poset of the standard representation V of A . The weight poset of S m ( V ) is { ( m − k ) ϕ + kϕ | k = 0 , , , . . . , m } . Thereby, the weight poset of ∧ S m ( V ) is { (2 m − k − k ) ϕ + ( k + k ) ϕ | ≤ k < k ≤ m } . If m ≥
3, thendim ∧ S m ( V ) (2 m − ϕ +3 ϕ = 2 for the cases ( k , k ) = (0 , , (1 , ∧ S m ( V )is not weight multiplicity-free and any irreducible representations S m ′ ( V ) of A areweight multiplicity-free, ∧ S m ( V ) is not irreducible. Hence S m ( V ) ( m ≥
3) is notdense. It is well-known that the first fundamental representations of C n and G arenot dense. (For C n , see Example 2.6 or [2, § G , see [2, § B with highest weight ω is not dense since it is equivalent tothe first fundamental representation for C (for C , see Example 2.6 or [2, § (cid:3) To simplify the classification of thick representations, we introduce the notion ofgeometric equivalence.
KAZUNORI NAKAMOTO AND YASUHIRO OMODA
Definition 3.8 ( cf. [1, § § . For two representations ρ : G → GL( V ) and ρ ′ : G ′ → GL( V ′ ), we say that they are geometrically equivalent if there exists a C -linear isomorphism f : V → V ′ such that for the induced isomorphism f ∗ : GL( V ) → GL( V ′ ) we have f ∗ ( ρ ( G )) = ρ ′ ( G ′ ). Example 3.9.
Let ρ ∗ : G → GL( V ∗ ) be the dual representation of ρ : G → GL( V ).Then ρ and ρ ∗ are geometrically equivalent. Remark 3.10.
Assume that two representations ρ : G → GL( V ) and ρ ′ : G ′ → GL( V ′ ) are geometric equivalent. Then ρ is thick (resp. dense) if and only if so is ρ ′ .According to [4, § G with highest weight ω by G . Similarly, S m G stands for the m -th symmetric power of G . In addition, let e denote the trivial 1-dimensionalrepresentation for any groups G . Then we have: Theorem 3.11.
If a representation of a connected simple Lie group is thick, thenit is geometrically equivalent to one of the following list: e, SL n ( n ≥ , S m SL ( m ≥ , SO n +1 ( n ≥ , Sp n ( n ≥ , G . If a representation of a connected simple Lie group is dense, then it is geometricallyequivalent to one of the following list: e, SL n ( n ≥ , S SL , SO n +1 ( n ≥ . Proof.
The last fundamental representation of B with highest weight ω is geometricequivalent to the first fundamental representation of C with highest weight ω , thatis, Sp . By Theorems 3.6 and 3.7, we have the classification above. (cid:3) Theorem 3.11 also shows the list of geometrically equivalences of thick (or dense)representations of connected semi-simple Lie groups.
Theorem 3.12.
Any thick representation V of a connected semi-simple Lie group G is geometrically equivalent to one of the list in Theorem 3.11. In particular, the listof geometrically equivalences of thick representations (resp. dense representations)of connected semi-simple Lie groups is the same as that of thick representations(resp. dense representations) of connected simple Lie groups in Theorem 3.11.Proof. Let ρ : G → GL( V ) be a thick representation of a connected semi-simpleLie group G . Take a universal cover π : e G = G × G × · · · × G r → G , where G i is a simply-connected simple Lie group for each i = 1 , , . . . , r . We have a thickrepresentation e ρ = ρ ◦ π : e G → GL( V ). Since V is an irreducible representationof e G , there exist irreducible representations V i of G i (1 ≤ i ≤ r ) such that V ∼ = V ⊗ V ⊗ · · · ⊗ V r as representations of e G . By Theorem 3.5, V is WMF as arepresentation of e G and the weight poset W e G ( V ) is a totally ordered set. Here, HE CLASSIFICATION OF THICK REPRESENTATIONS OF SIMPLE LIE GROUPS 9 weights in W e G ( V ) are with respect to a maximal torus T = T × T × · · · × T r of e G ,where T i is a maximal torus of G i . The order in W e G ( V ) is defined with respect to aset ∆ = ∆ ⊔ ∆ ⊔ · · · ⊔ ∆ r of simple roots of e G , where ∆ i is a set of simple roots of G i . Let W G i ( V i ) be the weight poset (with respect to T i and ∆ i ) of the G i -module V i . We can write W e G ( V ) = { P ri =1 ψ i | ψ i ∈ W G i ( V i ) } .Suppose that there exists 1 ≤ i < j ≤ r such that e ρ ( G i ) = { e } and e ρ ( G j ) = { e } .Then ♯W G i ( V i ) ≥ ♯W G j ( V j ) ≥
2. Choose φ , φ ∈ W G i ( V i ) and ϕ , ϕ ∈ W G j ( V j ) such that φ > φ and ϕ > ϕ . Let ξ = P k = i,j ψ k be the sum of the highestweights ψ k ∈ W G k ( V k ) for k = i, j . For η = ξ + φ + ϕ , η = ξ + φ + ϕ ∈ W e G ( V ),neither η > η nor η < η holds. This implies that W e G ( V ) is not totally ordered,which is a contradiction. Hence, any G k satisfy e ρ ( G k ) = { e } except some G i . Since V k = C except for k = i , the representation V of e G is geometrically equivalent tothe representation V i of G i . In particular, the representation V of G is geometricallyequivalent to a thick representation V i of a connected simple Lie group G i . Therefore,Theorem 3.11 also shows the lists of geometrically equivalences of thick and denserepresentations of connected semi-simple Lie groups. (cid:3) References [1] C. Benson and G. Ratcliff, On multiplicity free actions, Representations of real and p -adicgroups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 2, Singapore Univ. Press, Sin-gapore, (2004), 221–304.[2] W. Fulton and J. Harris, Representation theory. A first course , Graduate Texts in Mathemat-ics, 129, Springer-Verlag, New York, (1991).[3] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond,The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan,(1995) 1–182.[4] V. Kac, Some remarks on nilpotent orbits, J. Algebra (1980), no. 1, 190–213.[5] F. Knop, Some remarks on multiplicity free spaces, Representation theories and algebraicgeometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer Acad. Publ., Dordrecht,(1998), 301–317.[6] K. Nakamoto, Representation varieties and character varieties, Publ. Res. Inst. Math. Sci. (2000), no. 2, 159–189.[7] K. Nakamoto and Y. Omoda, Thick representations and dense representations I, Kodai Math.J. (2019), no. 2, 274–307.[8] D. Panyushev, Properties of weight posets for weight multiplicity free representations, Mosc.Math. J. (2009), no. 4, 867–883. Center for Medical Education and Sciences, Faculty of Medicine, University ofYamanashi
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