Real Representations of C 2 -Graded Groups: The Linear and Hermitian Theories
aa r X i v : . [ m a t h . R T ] A ug REAL REPRESENTATIONS OF C -GRADED GROUPS: THELINEAR AND HERMITIAN THEORIES DMITRIY RUMYNIN AND JAMES TAYLOR
Abstract.
We study linear and hermitian representations of finite C -gradedgroups. We prove that the category of linear representations is equivalent to acategory of antilinear representations as an ∞ -category. We also prove that thecategory hermitian representations, as an ∞ -category, is equivalent to a categoryof usual representations. Real representations first appear in Quantum Mechanics in the works of Wigner[15]. Independently, they are introduced by Atiyah and Segal [1] and Karoubi [6]in the context of equivariant KR-theory. Over the time they were actively studiedby many scientists with a range of backgrounds (cf. [2, 3, 4, 9, 10, 11, 13, 16]).The present paper is a sequel to our study of antilinear representations [12]. Herewe investigate linear and hermitian representations, introduced by Young [16].A C -graded group is a pair G ≤ b G , where G is an index 2 subgroup of b G .A Real representation of G is a complex representation ( V, ρ ) of G together with“an action” of the other coset b G \ G satisfying appropriate algebraic coherenceconditions. In the antilinear theory, each element w ∈ b G \ G acts by an antilinearoperator, or simply a linear map ρ ( w ) : V → V . In the linear theory, w ∈ b G \ G acts by a bilinear form, regarded as a linear map ρ ( w ) : V ∗ → V . Finally, in thehermitian theory, an element w ∈ b G \ G acts by a sesquilinear form, regarded asa linear map ρ ( w ) : V ∗ → V .The goal of the present paper is to describe the linear and hermitian categoriesfully. These categories are not R -linear. Instead they are topological. More-over, they are “homotopically equivalent” to some non-full subcategories of the Date : August 17, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Real representation, finite group, topological category, ∞ -category.The first author was partially supported within the framework of the HSE University BasicResearch Program and the Russian Academic Excellence Project ‘5–100’.We are indebted to Matthew B. Young for useful conversations and interest in our work. Wewould like to thank Karl-Hermann Neeb for bringing the work of Wigner to our attention. Weare grateful to John Jones for tutoring us in Homotopic Topology. categories of antilinear or usual representations. Let us now provide a detaileddescription of the content of the present paper.We start by reminding the reader the basic notions of ∞ -categories in Section 1.Then we introduce all the categories that we study in the present paper. We finishthe section with the statement of the main result of this paper, Theorem 1.5.In section 2 we prove the first two parts of the main theorem, describing the twoversions of the linear theory. The first statement reduces to homotopy equivalencesbetween certain Lie groups and their maximal compact subgroups (see Table 1).Similarly, the second statement to homotopy equivalences between homogeneousspaces.Section 3 is devoted to the proof the last two parts of the main theorem, de-scribing the hermitian theory. The proofs are parallel to the first two parts withdifferent Lie groups and homogeneous space appearing (see Table 3).Finally, in Section 4 we discuss some generalisations, outlining directions forfuture research. 1. Categories ∞ -Categories. Let T be the closed monoidal category of compactly gener-ated weakly Hausdorff topological spaces together with its Quillen model structure[14, App. A] (cf. [5]). A topological category is a category C , enriched in T .Given a topological space Z ∈ T , by [[ Z ]] we denote the corresponding object inthe homotopy category Ho( T ). The homotopy category Ho( T ) is closed monoidal.By [[ C ]] we denote the category with the same objects as C and new morphisms[[ C ]]( X, Y ) := [[ C ( X, Y )]] . Since the morphisms are not sets, it is not a category in the usual sense. Insteadit is an enriched in Ho( T ) category.In this paper by ∞ -categories we understand categories of the form [[ C ]], enrichedin Ho( T ), coming from topological categories. A functor (or an equivalence) of ∞ -categories is just a functor (an equivalence) of categories enriched in Ho( T ).This rather restrictive view of ∞ -categories , outlined by Lurie [7, Def. 1.1.1.6], issufficient for our ends.Let C be a topological category. By Mono( C ) and Iso( C ) we denote the monomor-phism and isomorphism categories of C . They have the same objects as C but fewermorphisms(1) Mono( C )( X, Y ) := k ( { f ∈ C ( X, Y ) | f is a monomorphism } ) , Iso( C )( X, Y ) := k ( { f ∈ C ( X, Y ) | f is an isomorphism } ) . A subset of C ( X, Y ), equipped with the subspace topology, is weakly Hausdorff[14, Prop A.4] but not necessarily compactly generated. Hence, we apply the
EAL REPRESENTATIONS 3 kellification functor k to the subspace: the closed subsets of k ( Z ) are compactlyclosed subsets of Z , i.e., those subsets A ⊆ Z that f − ( A ) is closed in K for anycompact K and any continuous map f : K → Z [14, A.1].Thus, both Mono( C ) and Iso( C ) are topological categories. The category Iso( C )is often called the core of the category C .1.2. Modules.
Let A be an associative algebra over R or C , Mod fd ( A ) its cate-gory of finite-dimensional modules. Each hom-set in Mod fd ( A ) is a finite-dimensionalvector space over R . Considering it in its Euclidean topology yields a topologicalcategory structure on Mod fd ( A ).By a C -graded group we understand an exact sequence of finite groups1 → G → b G π −→ C = {± } → . An element g ∈ b G is called even (odd) if π ( g ) = 1 ( π ( g ) = − C b G and the skew groupalgebra C ∗ b G , where the coset b G \ G acts on C by complex conjugation. We studythe corresponding topological categories: R ( G ) := Mod fd ( C b G ) and A ( G ) := Mod fd ( C ∗ b G ) . The category of R ( G ) is the category of representations of b G . Following on fromour previous work [12], we think of the C -graded group b G as a Real structure on G and then of A ( G ) as the category of antilinear Real representations of G , whichis the reason to keep G but not b G in the notation. Similarly to Section 1.1, weare interested in the categories of isomorphisms and monomorphisms of A ( G ) and R ( G ). Since the original categories are abelian, the monomorphisms are preciselythe injective maps and the isomorphisms are precisely the bijective maps. Notethat the kellification functor in (1) does not change the topology for the four newcategories. Indeed, the hom-set Mono( R ( G ))( X, Y ) is open in R ( G )( X, Y ), hence,first countable, while every first countable topological space is compactly generated[14, A.2]. Similarly for the other three topological categories.1.3.
Hermitian Representations Over a Ring.
Let K = ( K , ι ) be a commuta-tive ring with involution ι ( a ) = a , which is allowed to be trivial. Let Mod fgp ( K G )be the category of representations of G over K : we define it as the full subcat-egory of Mod ( K G ) consisting of objects that are finitely generated projective K -modules. EAL REPRESENTATIONS 4
We need adjectives describing sesquilinear forms (cf. [12, 2.7]). Let w ∈ b G \ G .A sesquilinear form B : V × V → K on a V ∈ Mod fgp ( K G ) is called(2) w -invariant if B ( g u, wgw − v ) = B ( u, v ) for all g ∈ G, u, v ∈ V , w -skew-hermitian if B ( u, w v ) = − B ( v, u ) for all u, v ∈ V , and w -hermitian if B ( u, w v ) = B ( v, u ) for all u, v ∈ V .
These properties do not depend on a particular choice of w : a w -invariant formis v -invariant for any v ∈ b G \ G , etc. If the involution is known to be trivial, weroutinely use the words w -symmetric and w -alternating instead of w -hermitianand w -skew-hermitian.By V ∗ we denote the dual module, by V – the conjugate module. Let ev : V → V ∗∗ = V ∗∗ , ev v ( f ) = f ( v ) be the canonical isomorphism of V with its double dual K -module. For a K -linear map f : V → W , we denote the conjugate-transposemap f ∗ = f ∗ : W ∗ → V ∗ . A convenient notation is( ǫ V, ǫ f ) := (cid:26) ( V, f ) if ǫ = 1 , ( V ∗ , f ∗ ) if ǫ = − , and δ x,y, − := (cid:26) x = y = − , . Definition 1.1. A hermitian K -representation of a C -graded group b G (or a Realgroup G ) is a finitely generated projective K -module V with invertible linear maps ρ ( z ) : π ( z ) V → V for all z ∈ ˆ G , such that ρ ( e ) = id V , and(3) ρ ( z z ) = ρ ( z ) ◦ π ( z ) ρ ( z ) π ( z ) ◦ ev δ π ( z ,π ( z , − . In other words, each odd element w defines a non-degenerate sesquilinear form B w : V × V → K , B w ( u, v ) := ρ ( w ) − ( v )( u ) . If V is a free K -module, we can choose a basis of V and write the bilinear formsas matrices: B w ( u, v ) = v T B ( w ) u for all u, v ∈ V , where u is the coordinate column of u . We can also write the linear maps ρ ( z ) asmatrices M ( z ), using the dual basis of V ∗ for odd w . Note that M ( w ) = B ( w ) ♭ where A ♭ := ( A − ) ∗ for an invertible matrix A . It is instructive to write (3) asfour different conditions depending on the parity of elements. The condition (3)for two even elements means that ( V, ρ ) is a representation of G . The other threecorners tell us thateven-odd odd-even odd-odd B gw ( u, v ) = B w ( u, g − v ) B wg ( u, v ) = B w ( g u, v ) B w ( u, v ) = B w (( w w ) − v, u ) B ( gw ) = M ( g ) ♭ B ( w ) B ( wg ) = B ( w ) M ( g ) M ( w w ) ♭ = B ( w ) B ( w ) ♭ M ( gw ) = M ( g ) M ( w ) M ( wg ) = M ( w ) M ( g ) ♭ M ( w w ) = M ( w ) M ( w ) ♭ EAL REPRESENTATIONS 5 for all g ∈ G, w , w , w ∈ b G \ G , u, v ∈ V . The following useful reformulation ofthese conditions is easy to prove: Lemma 1.2.
Suppose that V ∈ Mod fgp ( K G ) has a non-degenerate sesquilinearform B w for each w ∈ b G \ G . This data determines a hermitian K -representation ifand only if each B w is w -invariant and w -hermitian, and B w ( u, v ) = B w ( u, w w − v ) for all w , w ∈ b G \ G . There are two competing notions of a homomorphism. Let (
V, ρ ), (
W, µ ) be twohermitian K -representations of G . By a homomorphism of hermitian representa-tions we understand a homomorphism of K G -modules f : V → W that preservesall the forms B w : B V w ( u, v ) = B W w ( f ( u ) , f ( v )) for all u, v ∈ V, w ∈ b G \ G .
Thanks to the odd-odd corner condition, this is equivalent to preserving one ofthe forms B w . Since the forms B w are non-degenerate, all homomorphisms areinjective. We denote the resulting category H ( K ,ι ) ( G ).Now by a strong homomorphism we understand a K -linear map f : V → W such that the squares V WV W fρ ( g ) µ ( g ) f W ∗ V ∗ W V f ∗ µ ( z ) ρ ( z ) f commute for all g ∈ G and z ∈ b G \ G . Clearly, a homomorphism f is a strong ho-momorphism if and only if f preserves the dual forms B ∗ w . This is also equivalent to f being bijective. This is further equivalent to f being an isomorphism. Thus, thecategory of hermitian K -modules with strong isomorphisms is just Iso( H ( K ,ι ) ( G )).Observe that there is a natural notion of “direct sum” ( V, ρ ) ⊕ ( W, µ ) of hermitian K -representations: it is a direct sum of K G -modules with obvious extension of theforms B w ( v + w , v + w ) := B w ( v , v ) + B w ( w , w ). The quotation marks arejustified by the limited categorical properties of this construction: “the direct sum”is a coproduct (but not a product) in H ( K ,ι ) ( G ) and has no categorical propertiesin Iso( H ( K ,ι ) ( G )).1.4. Maschke’s Theorem.
Suppose that K is a field. The involution ι could stillbe trivial or non-trivial. This allows to take orthogonal complements V = W ⊕ W ⊥ on finite-dimensional modules with hermitian forms as soon as the restriction ofthe form to W is non-degenerate.Consider ( V, ρ ) ∈ H ( K ,ι ) ( G ). A vector subspace W ⊆ V is a subrepresentation,if it is a K G -submodule and all restrictions B V w | W are non-degenerate. These EAL REPRESENTATIONS 6 restrictions define a structure of hermitian representation on W such that theembedding W ֒ → V is a morphism in H ( K ,ι ) ( G ). Proposition 1.3. (Maschke’s Theorem) Suppose that K is a field and ( V, ρ ) ∈H ( K ,ι ) ( G ) . If ( W, µ ) is a subrepresentation of V , then the right orthogonal com-plement W ⊥ under B V w is a Hermitian subrepresentation with W ⊥ ⊕ W = V .Moreover, W ⊥ does not depend on the choice of odd w .Proof. Since B V w | W = B W w , W ∩ W ⊥ = 0. By non-degeneracy of B V w , W ⊕ W ⊥ = V .The odd-odd version of (3) implies that W ⊥ is independent of w . The even-oddversion of (3) implies that W ⊥ is a K G -submodule. Finally, we can equip W ⊥ with the action of odd elements by B W ⊥ w := B V w | W ⊥ . (cid:3) We say that V ∈ H ( K ,ι ) ( G ) is irreducible if V = 0 and 0, V are the onlysubrepresentations of V . Corollary 1.4. (Krull-Remak-Schmidt Theorem) Every V ∈ H ( K ,ι ) ( G ) decomposesas a finite direct sum of irreducible hermitian representations in a unique way upto permutations and isomorphisms.Proof. The decomposition easily follows from Proposition 1.3.Suppose that V = V ⊕ . . . ⊕ V m = W ⊕ . . . ⊕ W n are two decompositions.Uniqueness is proved by induction on m . One of the maps V ֒ → V ։ W j must bean isomorphism in H ( K ,ι ) ( G ). This is the induction base. Furthermore, this givesan isomorphism between V ⊥ and W ⊥ j , which is the induction step. (cid:3) Statement of the Main Theorem.
The most interesting field for us arethe complex numbers C . It has two natural involutions. The first involution is thecomplex conjugation. The corresponding category is denoted H ( G ) := H ( C ,ι ) ( G ).We call these representations hermitian or simply H-representations. We denotethe hom-sets in this category by Hom H ( V, W ) := H ( G )( V, W ) and Aut H ( V ) := H ( G )( V, V ).The second involution is trivial. The corresponding category is denoted L ( G ) := H ( C , Id) ( G ). We call these representations linear or simply L-representations. Wedenote the hom-sets in this category by Hom L ( V, W ) := L ( G )( V, W ) and Aut L ( V ) := L ( G )( V, V ). Theorem 1.5.
Let b G be a finite C -graded group b G . The following pairs of ∞ -categories are equivalent:(i) [[Iso( A ( G ))]] and [[Iso( L ( G ))]] ,(ii) [[Mono( A ( G ))]] and [[ L ( G )]] ,(iii) [[Iso( R ( G ))]] and [[Iso( H ( G ))]] ,(iv) [[Mono( R ( G ))]] and [[ H ( G )]] . EAL REPRESENTATIONS 7 The Linear Theory
Structure of L-Representations.
Let V be an irreducible L-representation.Each odd element w yields an isomorphism of C G -modules ρ ( w ) : V ∗ → w · V . Itfollows that the character χ of the underlying C G -module V satisfies χ = w · χ .The following result describes its structure. Proposition 2.1.
One of the following mutually exclusive statements holds for anirreducible L-representation V .(1) V ↓ C G = W is a simple C G -module; W ∼ = w · W as C G -modules; W is ofantilinear type R ; Aut L ( V ) = {± id } .(2) V ↓ C G = W ⊕ W ′ is the sum of two simple C G -modules, both of antilineartype C ; W = W ′ and W = w · W as C G -modules; Aut L ( V ) ∼ = C \ .(3) V ↓ C G = W ⊕ W ′ is the sum of two simple C G -modules, both of antilineartype H ; W ∼ = W ′ and W ∼ = w · W as C G -modules; Aut L ( V ) ∼ = SL ( C ) .Proof. Let W be a simple C G -submodule of V . Since the form B w is w -invariant, W ⊥ is a C G -submodule of V . Because the form is w -symmetric, W ⊥ = ⊥ W andker( B w | W ) = W ⊥ ∩ W is a C G -submodule of W . It must be zero or W . Hence,we have two mutually exclusive cases. Case A: Some simple C G -submodule W of V satisfies W ⊥ ∩ W = 0 . Itfollows that W is an L-subrepresentation of V , hence, W = V . By [12, Prop.2.15], V has antilinear type R and the bilinear form B w yields an isomorphism W ∼ = w · W .Since W is simple, any C G -automorphism of V is a scalar α id, α ∈ C \ { } . Theonly scalars preserving the bilinear form B w are ±
1, hence, Aut L ( V ) = {± id } . This is statement (1).
Case B: Any simple C G -submodule W of V satisfies W ⊥ ∩ W = W . Fix W . Notice that W ⊂ W ⊥ and write V = W ⊥ ⊕ W ′ as a C G -module.Observe that W ⊕ W ′ is an L-subrepresentation. Indeed,ker( B w | W ⊕ W ′ ) = ( W ⊕ W ′ ) ⊥ ∩ ( W ⊕ W ′ ) ⊆ W ⊥ ∩ ( W ⊕ W ′ ) = W and, by the non-degeneracy of B w on V , ker( B w | W ⊕ W ′ ) = 0. By the irreducibilityof V , V = W ⊕ W ′ and, therefore, W = W ⊥ .It follows that 2 dim W = dim W + dim W ⊥ = dim V . Moreover, W ′ is alsoa simple C G -module because any simple C G -submodule U of W ′ also satisfies2 dim U = dim V .The C G -module homomorphism between simple C G -modules defined by B w (4) g : w · W ֒ → w · V f −→ V ∗ ։ W ′∗ , f ( u )( v ) = B w ( v, u ) EAL REPRESENTATIONS 8 is non-zero because W = W ⊥ . Furthermore, g determines B w because B w is w -symmetric: B w ( u + u ′ , v + v ′ ) = B w ( u, v ′ ) + B w ( u ′ , v ) = g ( u )( w − v ′ ) + g ( v )( u ′ )(5)for all u + u ′ , v + v ′ ∈ W ⊕ W ′ . We have two further subcases. Subcase B1: W = W ′ . Then W = w · W , so W is of antilinear type C [12,Thm. 5.4]. Let f : V → V be a morphism of L-representations, A – the matrixof f . Note that B ( w ) = A T B ( w ) A for a fixed w is a necessary and sufficientcondition for f ∈ Aut C G ( V ) to be a morphism of L-representations. As W = W ′ and W = W ⊥ , A and B ( w ) have the form B ( w ) = (cid:18) YX (cid:19) , A = (cid:18) α id β id (cid:19) , thus (cid:18) YX (cid:19) = (cid:18) αβYαβX (cid:19) . It follows that αβ = 1 and Aut L ( V ) ∼ = C \ This is statement (2).
Subcase B2: W ∼ = W ′ . Then W ∼ = w · W . To show that W is of antilinear type H , it suffices to construct a non-degenerate, w -invariant, w -alternating bilinearform on W [12, Thm. 5.4].Let h : W → W ′ be a C G -module isomorphism. Consider e V := W ⊕ W . Letus use the C G -module isomorphism id ⊕ h : e V → V to turn e V into an irreducibleL-representation: e B w (( u , u ) , ( v , v )) := B w ( u + h ( u ) , v + h ( v )) . Consider isomorphisms of C G -modules f , f : w · W → W ∗ where(6) f ( w )( v ) = e B w ((0 , v ) , ( w, , f ( w )( v ) = e B w (( v, , (0 , w )) . There exists λ ∈ C such that f = λf . As the form is w -symmetric, f ( w )( v ) = f ( w v )( w ). Then f ( w )( v ) = λf ( w )( v ) = λf ( w v )( w ) = λ f ( w v )( w )= λ f ( w w )( w v ) = λ f ( w )( v ) . Therefore, λ = ±
1. If λ = 1, then the diagonal { ( w, w ) } is an L-subrepresentationof e V , which contradicts irreducibility. Thus, λ = − D ( u, v ) := f ( v )( u ) on W is non-degenerate, w -invariant and w -alternating. Thus, W isof antilinear type H . Moreover, this shows that e B w is symplectic. As before, B ( w ) = A T B ( w ) A is a necessary and sufficient condition for f ∈ Aut C G ( V ) to bea morphism of L-representations. Now A and B ( w ) have the form B ( w ) = (cid:18) − XX (cid:19) , A = (cid:18) α id β id γ id δ id (cid:19) , thus (cid:18) − XX (cid:19) = ( αδ − βγ ) (cid:18) − XX (cid:19) . EAL REPRESENTATIONS 9
Thus, A is an L-homomorphism if and only if (cid:0) α βγ δ (cid:1) ∈ SL ( C ). Therefore, Aut L ( V ) ∼ =SL ( C ). This is statement (3). (cid:3)
We can now describe all L-representations.
Corollary 2.2.
Any L-representation is determined up to isomorphism by theunderlying C G -module.Proof. By Maschke’s Theorem (Proposition 1.3) and Krull-Remak-Schmidt Theo-rem (Corollary 1.4), it suffices to prove the corollary for irreducible L-representations.Let V be an irreducible L -representation, whose C G -module carries a second L-representation structure, denoted e V and e B w . For V , consider the three mutuallyexclusive cases in Proposition 2.1.In case (1), e B w = αB w for some α ∈ C , by Schur Lemma. Any choice of thesquare root yields an isomorphism of L-representations √ α id : e V → V .In the cases (2) and (3), the orthogonal complements under e B w and αB w areinevitably equal: W ′ = f W ′ . Moreover, the maps in (4) are scalar multiples: e g w = αg w for some α ∈ C \
0. It follows from (5) that e B w = (cid:18) α id α id (cid:19) B w and (cid:18) √ α id √ α id (cid:19) : e V → V is an isomorphism of L-representations. (cid:3) The three mutually exclusive possibilities for an irreducible L-representation inProposition 2.1 correspond exactly to antilinear types [12, Table 4]. In particular,any L-representation is an A-representation.Now pick an A-representation V . Using the unitary trick, fix a G -invarianthermitian form h· , ·i on it. Define(7) B w ( u, v ) := h u, w − v i + h v, w u i . Lemma 2.3.
Formula (7) defines an L-representation structure on V .Proof. The forms are non-degenerate. Consider u ∈ V such that B w ( u, v ) = 0 forall v ∈ V . Hence, 0 = B w ( u, w u ) = h u, u i + h w u, w u i . Since h w, w i ≥ w ∈ V , we conclude that h u, u i = 0 and u = 0.It remains to verify the conditions of Lemma 1.2. B w is w -invariant: B w ( g u, wgw − v ) = h g u, gw − v i + h wgw − v, wg u i = h u, w − v i + h v, w u i = B w ( u, v ) . The second equality holds by G -invariance. B w is w -symmetric: B w ( u, w v ) = h u, w v i + h w v, w u i = h u, w v i + h v, w − u i = B w ( v, u ) . EAL REPRESENTATIONS 10
The final condition holds as well: B w ( u, w w − v ) = h u, w − v i + h w w − v, w u i = h u, w − v i + h v, w u i = B w ( u, v ) . (cid:3) Let us summarise the discussion in this section with the next result.
Corollary 2.4. A C G -module extends to an A-representation if and only if itextends to an L-representation. This gives a bijection between isomorphism classesof A-representations and L-representations. Intermediate Category and Functors.
Let V , . . . , V k be a complete set ofdistinct irreducible A-representations. We turn them into irreducible L-representationsas in Lemma 2.3, by choosing a G -invariant hermitian form h· , ·i on each V i .Let us consider the skeletons of L ( G ) and A ( G ) that consist of all finite di-rect sums ⊕ ki =1 n i V i . Each object in this skeleton is canonically (after previouschoices) equipped with a G -invariant hermitian form. Let use define the interme-diate category A ∗ ( G ) on this skeleton: A ∗ ( G )( V, W ) consist of those morphisms in A ( G )( V, W ) that preserve the canonical hermitian form. Such morphisms are nec-essarily injective. It follows from (7) that any morphism in A ∗ ( G )( V, W ) preservesall B w , hence A ∗ ( G )( V, W ) is a subset of L ( G )( V, W ) as well.Both inclusions define the same subspace topology on A ∗ ( G )( V, W ). Thus, wehave the following topological (enriched in T ) functors(8) L ( G ) Φ ←− A ∗ ( G ) Ψ −→ Mono( A ( G )) . Both functors are essentially surjective on objects. Hence, to prove parts (i) and(ii) of Theorem 1.5, it suffices to show that the functors are homotopy equivalenceson morphisms.2.3.
Proof of Part (i) of Theorem 1.5.
Since all morphisms are isomorphismsthe task is to compute endomorphisms of an object U = ⊕ i n i V i in all three cate-gories. The direct sum decomposition works in C G -modules. Hence,Aut L ( U ) = Y i Aut L ( n i V i ) , Aut A ( U ) = Y i Aut A ( n i V i )and Aut ∗ ( U ) := A ∗ ( G )( U, U ) = Y i A ∗ ( G )( n i V i , n i V i )with similar direct product decompositions for the functors: Φ( U ) = Q i Φ( n i V i )and Ψ( U ) = Q i Ψ( n i V i ). This reduces the theorem to the case of isotypical repre-sentation, that is, U = nV .It suffices to show that both Φ( nV ) and Ψ( nV ) are homotopy equivalencesfor an irreducible L-representation V of dimension d . These maps are homotopyequivalences between a Lie group and its maximal compact subgroup. The proof EAL REPRESENTATIONS 11
Table 1.
Homotopy Equivalences for Part (i)Type of V Aut L ( nV ) Aut ∗ ( nV ) Aut A ( nV ) R O n ( C ) O n ( R ) GL n ( R ) C GL n ( C ) U n ( C ) GL n ( C ) H Sp n ( C ) Sp( n ) GL n ( H )consists of computations of these groups: all algebraic homomorphisms in the proofare continuous. The result is summarised in Table 1. Case 1:
Let V be an irreducible A-representation of type R . This correspondsto Aut A ( V ) = R and the case (1) in Proposition 2.1. Then Aut A ( nV ) = GL n ( R ).The automorphisms in A ∗ ( G ) preserve a hermitian form so that Aut ∗ ( nV ) =GL n ( R ) ∩ U( nV ). Choose an orthonormal basis of V . Extend it by repeating toan orthonormal basis of nV . In this basis, U( nV ) consists of unitary matrices,while Aut A ( nV ) consist of the dn × dn -matrices M = ( M i,j ) n × n , where each block M i,j is α i,j id d , α i,j ∈ R . It follows that Aut ∗ ( nV ) = O n ( R ) is a maximal compactsubgroup of Aut A ( nV ) = GL n ( R ). The groups have two components and Ψ( nV ) isa homotopy equivalence between a Lie group and its maximal compact subgroup.The group Aut L ( nV ) is a subgroup of Aut C G ( nV ) = GL n ( C ). It consists of M ∈ GL n ( C ) ⊆ GL dn ( C ) subject to the extra condition(9) M T B nV ( w ) M = B nV ( w ) , which could be checked for one element w . In the basis as above, the matrix B nV ( w ) is block-diagonal with n square blocks B V ( w ). On the other hand, M ∈ Aut C G ( nV ) is of a special block structure as well: M = ( M i,j ) n × n , where M i,j = α i,j id d , α i,j ∈ C . Thus, condition (9) becomes ( α i,j ) T ( α i,j ) = id n and Aut L ( nV ) =O n ( C ). Hence, Φ( nV ) is a homotopy equivalence as well. Case 2:
Let V be an irreducible A-representation of type C . This correspondsto Aut A ( V ) = C and the case (2) in Proposition 2.1. In particular, V = W ⊕ W ′ as a C G -module and Aut A ( nV ) = GL n ( C ).To proceed, we choose an orthonormal bases of W and W ′ and replicate themthrough nV . This yields an explicit isomorphismAut C G ( nV ) → GL n ( C ) × GL n ( C ) , M = ( M i,j ) n × n (cid:16) ( α i,j ) n × n , ( β i,j ) n × n (cid:17) , where each block is a d × d -matrix of the form(10) M i,j = (cid:18) α ij id d/ d/ d/ β ij id d/ (cid:19) , α ij , β ij ∈ C . The subgroup Aut A ( nV ) consist of the matrices M with α ij = β ij for all i and j . Preservation of the hermitian form on nV is equivalent to M being hermitian, EAL REPRESENTATIONS 12 which, in turn, is equivalent to ( α ij ) n × n being hermitian. Thus, Aut ∗ ( nV ) = U n ( C )and Ψ( nV ) is a homotopy equivalence between a connected Lie group and itsmaximal compact subgroup.Let us consider M ∈ Aut L ( nV ) ≤ Aut C G ( nV ). It must be in the form (10),additionally satisfying the condition (9). In our basis as above, the matrix B nV ( w )is block-diagonal with n square blocks B V ( w ) = (cid:18) d/ XY d/ (cid:19) where X and Y are some invertible matrices. Thus, the condition (9) becomes( α i,j )( β i,j ) T = id n . It follows that M ( α i,j ) is an isomorphism Aut L ( nV ) ∼ =GL n ( C ) and Φ( nV ) is a homotopy equivalence. Case 3:
Let (
V, ρ ) be an irreducible A-representation of type H . This cor-responds to Aut A ( V ) = H and the case (3) in Proposition 2.1. In particular, V = W ⊕ W as a C G -module and Aut A ( nV ) = GL n ( H ).To proceed, we choose an orthonormal basis e , . . . e d/ of W . We choose w W asthe second direct summand in W and use w e , . . . w e d/ as an orthonormal basisthere. We replicate this basis of V through nV . This yields an explicit isomorphismAut C G ( nV ) → GL n ( C ) , M = ( M i,j ) n × n (cid:16) (cid:18) α ij β ij γ ij δ ij (cid:19) × (cid:17) n × n , where each block is a d × d -matrix of the form(11) M i,j = (cid:18) α ij id d/ β ij id d/ γ ij id d/ δ ij id d/ (cid:19) , α ij , β ij , γ ij , δ ij ∈ C . The subgroup Aut A ( nV ) consist of such matrices M that (cid:16) α ij β ij γ ij δ ij (cid:17) belongs toa fixed copy of quaternions H inside M ( C ) for each block M i,j . In particular,Aut A ( nV ) ∼ = GL n ( H ).Preservation of the hermitian form on nV is equivalent to M being hermit-ian. Since ( M i,j ) ∗ = ( M ∗ j,i ) and M ∗ i,j is quaternionic conjugation, this is equiv-alent to ( M ij ) n × n being hyperunitary, considered as quaternionic matrix. Thus,Aut ∗ ( nV ) = Sp( n ) and Ψ( nV ) is a homotopy equivalence between a connected Liegroup and its maximal compact subgroup.As shown in the proof of Proposition 2.1, the bilinear form B V w on the L-representation is skew-symmetric. Thus, B nV w is non-degenerate and also skew-symmetric. Hence, after identifying Aut C G ( V n ) with GL n ( C ), we get an isomor-phism Aut L ( nV ) ∼ = Sp n ( C ). Thus, Φ( nV ) is a homotopy equivalence. (cid:3) Proof of Part (ii) of Theorem 1.5.
The proof is a natural continuationof the proof in Section 2.3. It reduces to the case of two isotypical representations
EAL REPRESENTATIONS 13 nV and kV , k > n . Indeed, if k < n there are no morphisms, while k = n is donein Section 2.3. From the functors (8) we get embeddings of the hom-spaces(12) L ( G )( nV, kV ) Φ( nV,kV ) ←−−−−− A ∗ ( G )( nV, kV ) Ψ( nV,kV ) −−−−−→ Mono( A ( G ))( nV, kV ) . The spaces admit the natural actions of the automorphisms groups by compo-sitions on the right and on the left, for instance,Aut ∗ ( kV ) × A ∗ ( G )( nV, kV ) × Aut ∗ ( nV ) → A ∗ ( G )( nV, kV ) . By inspection, we will see that the spaces are homogeneous over Aut ∗ ( kV ). Tocomplete the proof, we need to compute them explicitly.Let K ∈ { R , C , H } , V – an irreducible A-representation of type K . ThenMono( A ( G ))( nV, kV )) is the space of injective K -linear maps K n → K k . Replicat-ing an orthonormal basis of V to nV (as in Section 2.3) identifies A ∗ ( G )( nV, kV )with the K -Stiefel manifold. The standard argument, based on the Gram-Schmidtprocess, proves that Ψ( nV, kV ) is a homotopy equivalence (see also Lemma 2.5).The second map Φ( nV, kV ) requires case-by-case considerations. Case 1:
Let V be an irreducible A-representation of type R . Proceeding sim-ilarly to the case 1 in Section 2.3, we identify L ( G )( nV, kV ) with the space of k × n -matrices over C with orthogonal columns. By Witt’s Extension Theorem, L ( G )( nV, kV ) ∼ = O k ( C ) / O n ( C ).Since n < k , we can restrict to matrices with the determinant 1 so that the mapΦ( nV, kV ) becomes the natural embedding of the homogeneous spacesSO k ( R ) / SO n ( R ) ∼ = A ∗ ( G )( nV, kV ) Φ( nV,kV ) −−−−−→ L ( G )( nV, kV ) ∼ = SO k ( C ) / SO n ( C ) . It is a homotopy equivalence by the following standard fact, whose proof is similarto [5, Theorem 4.14]
Lemma 2.5.
Suppose that H ⊆ G are connected Lie groups and H is a closedsubgroup. If H c ⊆ G c are maximal compact subgroups of H and G , then thenatural map G c /H c → G/H is a homotopy equivalence.Proof.
The commutative diagram of pointed connected topological spaces( H c , e ) ( G c , e ) ( G c /H c , eH c )( H, e ) (
G, e ) (
G/H, eH ) j H f c j G h c jf h contains fibrations h and h c . Hence, it yields a map of long exact sequences ofhomotopy groups EAL REPRESENTATIONS 14 · · · → π n ( H c ) π n ( G c ) π n ( G c /H c ) π n − ( H c ) → · · ·· · · → π n ( H, e ) π n ( G ) π n ( G/H ) π n − ( H ) → · · · π n ( j H ) π n ( f c ) π n ( j G ) π n ( h c ) π n ( j ) ̟ c π n − ( j H ) π n ( f ) π n ( h ) ̟ where all π n ( j H ) and π n ( j G ) are isomorphisms. It follows that π n ( j ) are isomor-phisms and j is a weak homotopy equivalence. By Whitehead’s Theorem, j is ahomotopy equivalence. (cid:3) Case 2:
Let V be an irreducible A-representation of type C . Proceeding sim-ilarly to the case 2 in Section 2.3, we identify L ( G )( nV, kV ) with the space of k × n -matrices over C with unitary columns so that the maps Φ( nV, kV ) andΨ( nV, kV ) are the same natural embeddings of the complex Stiefel manifoldsU k ( C ) / U n ( C ) ∼ = A ∗ ( G )( nV, kV ) Φ( nV,kV ) −−−−−→ L ( G )( nV, kV ) ∼ = GL k ( C ) / GL n ( C ) . Case 3:
Let V be an irreducible A-representation of type H . Proceeding sim-ilarly to the case 3 in Section 2.3, we identify L ( G )( nV, kV ) with the space of2 k × n -matrices over C whose columns form a Darboux basis of a subspace. By(Symplectic) Witt’s Extension Theorem, L ( G )( nV, kV ) ∼ = Sp k ( C ) / Sp n ( C ). Themap Φ( nV, kV ) becomes the natural embedding of the homogeneous spaces, whichis a homotopy equivalence by Lemma 2.5:Sp( k ) / Sp( n ) ∼ = A ∗ ( G )( nV, kV ) Φ( nV,kV ) −−−−−→ L ( G )( nV, kV ) ∼ = Sp k ( C ) / Sp n ( C ) . (cid:3) The Hermitian Theory
Irreducible H-Representations.
Let V be an irreducible H-representation.Each odd element w yields an isomorphism of C G -modules ρ ( w ) : V ∗ → w · V ,thus, the character χ of the underlying C G -module V satisfies χ = w · χ .The following Proposition 3.1 describing the structure of V has one less case thanthe corresponding Proposition 2.1 for the linear theory. This essential difference isdue to the fact that w -invariant bilinear and sesquilinear forms behave differentlyunder scaling.Consider a w -invariant form bilinear B on a C G -module W . Then necessarily B ( u, w v ) = λB ( v, u ) for λ ∈ {± } , and for any r ∈ C , the bilinear form e B := rB also satisfies e B ( u, w v ) = λ e B ( v, u ). Conversely, if B is sesquilinear, then B ( u, w v ) = λB ( v, u ) for λ ∈ S . Then for r ∈ C , e B := rB satisfies instead e B ( u, w v ) = rλB ( v, u ) = rr λ e B ( v, u ) . EAL REPRESENTATIONS 15
Table 2.
Index 2 Induction and Restriction V ↓ W W ⊕ w · WW ↑ V ⊕ ( V ⊗ π ) VW ∼ = w · W ? Yes No V ∼ = V ⊗ π ? No YesIn particular, as λ ∈ S , we can always choose r with r = λ − to make e B w -symmetric. Proposition 3.1.
Let V be an irreducible H-representation. One of the followingmutually exclusive statements hold.(1) W := V ↓ C G is a simple C G -module; W ∼ = w · W as C G -modules; Aut H ( V ) = { λ id | | λ | = 1 } .(2) V ↓ C G = W ⊕ W ′ decomposes as the sum of two simple C G -modules; W = W ′ and W = w · W as C G -modules; Aut H ( V ) ∼ = C \ .Proof. The proof of 2.1 remains true mutatis mutandis. The only major change isthat when V is the direct sum of two simple modules, V = W ⊕ W ′ , necessarily W = W ′ .Indeed, suppose h : W ∼ = −→ W ′ is an isomorphism. Construct an irreducible H-representation on e V := W ⊕ W by transferring the structure via the C G -moduleisomorphism id ⊕ h : e V → V . We have an isomorphism of H-representations as in(6) with f = λf for some λ ∈ S ⊆ C . Choose a square root η ∈ S , η = λ .Then h ( ηw, w ) i C is an H-subrepresentation of e V , a contradiction. (cid:3) Unlike the linear theory, the underlying C G -module does not always determinethe H-representation. Corollary 3.2.
Let V be an irreducible H-representation. If V is of type (1), thenthere are two non-isomorphic H-representations on V ↓ C G : these are ( V, B w ) and ( V, − B w ) . If V is of type (2), then V is determined by V ↓ C G . Relationship with C b G -Modules. Table 2 summarises the relationship be-tween simple modules of C G and C b G . There V is a simple C b G -module; W is asimple submodule of V ↓ C G . Corollary 3.3. A C G -module is extendible to a C b G -module if and only if it isextendible to an H-representation. This gives a bijection between isomorphismclasses of C b G -modules and H-representations.Proof. The bijection is given by the formula (7). Note that if a C b G -module V defines ( V, B w ), then V ⊗ π defines ( V, − B w ). Finally, use Corollary 3.2. (cid:3) EAL REPRESENTATIONS 16
Table 3.
Homotopy Equivalences for Part (iii) U Aut H ( U ) R ∗ ( G )( U, U ) Aut R ( U ) nV i ⊕ m ( V i ⊗ π ) U n,m ( C ) U n ( C ) × U m ( C ) GL n ( C ) × GL m ( C ) nU i GL n ( C ) U n ( C ) GL n ( C )Notice that the bijection in Corollary 3.3 is natural in type (2) but not in type (1).In type (1), ( V, B w ) can correspond to either V or V ⊗ π . Thus, we can talk abouta natural C p -torsor of bijections.3.3. Intermediate Category and Functors.
Fix representatives of each iso-morphism class of irreducible C b G -modules: V , V ⊗ π , . . . , V p , V p ⊗ π , U , ... , U q .Here U i ⊗ π ∼ = U i and V i ⊗ π = V i .We turn these into irreducible H-representations as in Lemma 2.3, by choosing a G -invariant hermitian form h· , ·i on each. As in Section 2.2, consider the skeletonsof H ( G ) and R ( G ) consisting of all finite direct sums of the fixed irreduciblemodules. Define the intermediate category R ∗ ( G ) on this skeleton: morphisms in R ∗ ( G ) are those morphisms in R ( G ) that preserve the chosen hermitian form.We have the following topological (enriched in T ) functors(13) H ( G ) Φ ←− R ∗ ( G ) Ψ −→ Mono( R ( G )) , both of which are essentially surjective on objects. Hence, as before, to prove parts(iii) and (iv) of Theorem 1.5, it suffices to show that the functors are homotopyequivalences on morphisms.3.4. Proof of Part (iii) of Theorem 1.5.
In the proof of part (i) it sufficedto establish both homotopy equivalences for nV , where V is an irreducible L-representation. Here, in light of 3.2, we instead need to show that both Φ( U ) andΨ( U ) are homotopy equivalences for U = nU i or U = nV i ⊕ m ( V i ⊗ π ), as thereare no C G -morphisms between such U for distinct U i or V i .The result is summarised in Table 3: Φ( U ) and Ψ( U ) are always homotopyequivalences between a connected Lie group and its maximal compact subgroup. Case 1:
Let U = nV i ⊕ m ( V i ⊗ π ). The final two columns are standard classicalresults. Now let us show that Aut H ( V ) = U n,m ( C ).The group Aut H ( U ) is a subgroup of Aut C G ( U ) = GL n ( C ) × GL m ( C ). It consistsof M ∈ GL n ( C ) × GL m ( C ) ⊆ GL dn ( C ) × GL dm ( C ) subject to the extra condition (9)as before, as M ∈ Aut C G ( U ) has M = ( M i,j ) ( n + m ) × ( n + m ) , where M i,j = α i,j id d , α i,j ∈ C .In the basis as above, the matrix B U ( w ) is block-diagonal with n square blocks: B U ( w ) = diag( B V i ( w ) , ..., B V i ( w ) , − B V i ( w ) , ..., − B V i ( w )) = ( I n ⊕− I m )diag( B V i ( w )).Then B U ( w ) M = ( I n ⊕ − I m ) diag( B V i ( w )) M = ( I n ⊕ − I m ) M diag( B V i ( w )), hence EAL REPRESENTATIONS 17 the condition (9) is equivalent to M ∈ U n,m ( C ). Therefore, Φ( U ) is a homotopyequivalence. Case 2: U = nU i . The proof is identical to that of the case 2 of the proof ofpart (i), with appropriate transposes changed to conjugate-transposes. (cid:3) Proof of Part (iv) of Theorem 1.5.
This is a continuation of the previoussection. The proof reduces to showing that the embeddings of hom-spaces H ( G )( U, U ′ ) Φ( U,U ′ ) ←−−−− R ∗ ( G )( U, U ′ ) Ψ( U,U ′ ) −−−−→ Mono( R ( G ))( U, U ′ )are homotopy equivalences, where ( U, U ′ ) are in the following two cases. Case 1: ( U, U ′ ) = ( nV i ⊕ m ( V i ⊗ π ) , n ′ V i ⊕ m ′ ( V i ⊗ π )) for n ≤ n ′ , m ≤ m ′ .Replicating an orthonormal basis of V i to nV i and V i ⊗ π to m ( V i ⊗ π ) identi-fies R ∗ ( G )( U, U ′ ) on each isotypical component with a product of complex Stiefelmanifolds. The space Mono( R ( G ))( U, U ′ ) identifies with the product of the spacesof injective map C n → C n ′ and C m → C m ′ . By the Gram-Schmidt argument,Ψ( U, U ′ ) is a homotopy equivalence.Similarly to the case 3 of Section 2.4, Φ( U, U ′ ) is the natural embedding ofhomogeneous spaces, hence, a homotopy equivalence by Lemma 2.5: R ∗ ( G )( U, U ′ ) H ( G )( nV, kV )U n ′ ( C ) × U m ′ ( C ) / U n ( C ) × U m ( C ) U n ′ ,m ′ ( C ) / U n,m ( C ) . Φ( U,U ′ ) ∼ = ∼ = Case 2: ( U, U ′ ) = ( nU i , mU i ) for n < m . This is identical to the case 2 ofSection 2.4: both Ψ( U, U ′ ) and Φ( U, U ′ ) are (different) natural embeddings ofcomplex Stiefel manifolds. (cid:3) Generalisations
Compact Groups.
The antilinear theory works equally well for a compactgroup G instead of a finite group [9]. In particular, all the results of our earlierpaper [12] remain valid under these assumptions. Since irreducible continuousrepresentations of compact groups are finite-dimensional, the results of the presentpaper remain valid as well.4.2. Infinite Dimensional Spaces.
It is subtle to replace a compact group witha locally compact group. The antilinear irreducible representations are no longerfinite-dimensional. On the other hand, the linear and hermitian theory require astronger form of duality, available only for finite-dimensional vector spaces. Oneway around it is to consider Hilbert spaces and unitary representations instead.
EAL REPRESENTATIONS 18
It is interesting to investigate which of the results of the present paper would stillhold in this case.4.3.
General Coefficients.
Another interesting project, worth further attention,is to replace C with a more general field or even a ring, as we have done inSection 1.3. The antilinear theory (at least over a field) is probably attainable[12], although it will depend on a classification of graded division rings. Theformulation of the main results of the present paper will require some version ofhomotopy theory of schemes (cf. [8]).4.4. General Gradings.
Suppose that G ≤ b G is a more general grading suchthat the quotient b G/G is a Galois group of a field extension F ≤ K . The antilineartheory in this set-up is clear: these are just representations of the skew group ring K ∗ b G . It would be still interesting to develop the theory in full, in the spirit of[12]. It is not clear to us how to approach the linear and hermitian theories in thiscontext. References [1] M. Atiyah and G. Segal. Equivariant K -theory and completion. J. Differential Geometry ,3:1–18, 1969.[2] Z. Boreviˇc and G. Devjatko. On a Frobenius-Moore formula.
Algebra and Number Theory(Ordzhonikidze) , 3:3–8, 1978. (Russian).[3] F. Dyson. The threefold way. Algebraic structure of symmetry groups and ensembles inquantum mechanics.
J. Mathematical Phys. , 3:1199–1215, 1962.[4] C.-K. Fok. The Real K -theory of compact Lie groups. SIGMA Symmetry Integrability Geom.Methods Appl. , 10:Paper 022, 26, 2014.[5] K. Hristova, J. Jones, and D Rumynin. General comodule-contramodule correspondence.arXiv:2004.12953, 2020.[6] M. Karoubi. Sur la K -th´eorie ´equivariante. In S´eminaire Heidelberg-Saarbr¨ucken-Strasbourgsur la K-th´eorie (1967/68) , LNM, Vol. 136, pages 187–253. Springer, 1970.[7] J. Lurie.
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Banach Cen-ter Publications, Volume 113 , pages 291–362, 2017.[10] B. Noohi and M. Young. Twisted loop transgression and higher jandl gerbes over finitegroupoids. arXiv:1910.01422, 2019.[11] K. Rozenbaum. Induced symplectic modules.
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EAL REPRESENTATIONS 19 [14] S. Schwede.
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