Distinguished orbits and the L-S category of simply connected compact Lie groups

DISTINGUISHED ORBITS AND THE L-S CA TEGOR Y OF SIMPL Y CONNECTE D COMP A CT LIE GR OUPS MARKUS HUNZIK ER AND MARK R. SEP ANS KI Abstra ct. W e show that the Lustern ik-Schnirelmann category of a simple, simply con- nected, compact Lie group G is b ounded abov e b y the sum of the relative categories of certain distinguished conjugacy classes in G corresp onding to the v ertices of the fun da- mental alco ve for the action of the affine W eyl group on the Lie algebra of a maximal torus of G . 1. In tro duction 1.1. The (n orm alized) Lusternik-Schnir elmann c ate g ory of a top ological space X , denoted cat( X ), is the le ast in teger m suc h that X can b e cov ered by m + 1 op en sets that are con tractible in X . On e of the p r oblems on Ganea’s list ([3]) from 1971 asks to fi nd the L-S catego ry of (compact) Lie groups. In 1975, Singhof ([9]) prov ed that cat (SU( n + 1)) = n . F or th e other families of simp ly conn ected compact Lie groups, the answ er is only known when the rank is small (cf. [7] for a nice su mmary of what is kno wn for simply connected and n on-simply connected compact Lie groups of small rank .) 1.2. The pur p ose of this sh ort note is to sh o w that the L -S cat egory of a simp le, simply connected, compact Lie group G is b ounded ab ov e b y the sum of the rela tive cat egories of certain d istinguished conju gacy classes in G . More precisely , sup p ose { v 0 , . . . , v n } are the v ertices of the fund amen tal alco v e for the action of the affine W eyl group on the Lie algebra of a maximal torus of G . F or 0 ≤ k ≤ n , let O k b e the conjugacy class of exp v k in G . Then w e will sh o w in Section 4 that cat( G ) + 1 ≤ n X k =0 (cat G ( O k ) + 1) , where cat G ( O k ) is the r elative L-S c ate gory of O k in G . (If Y ⊆ X is a top ologica l subsp ace, cat X ( Y ) is the least intege r m such that there there is a co ve r ing of Y by m + 1 op en subsets of X , eac h contract ible in X .) Date : Octob er 29, 2018. 1 2 MARKUS HUNZIKER AND MARK R. SEP ANSKI 1.3. F or G = SU( n + 1), the conju gacy classes O k turn out to b e the p oin ts of the center of G and we reco ver Singhof ’s result that cat(SU( n + 1)) ≤ n . F or G = Sp( n ), we conjecture that cat G ( O k ) ≤ min { k , n − k } (with resp ect to an app ropriate num b er in g) whic h would imply th at cat(Sp( n )) ≤  ( n + 2) 2 4  − 1 . Th u s for n = 1 , 2 , 3 , 4 , 5 , 6 , etc. our conjectured u pp er b oun d is 1 , 3 , 5 , 8 , 11 , 15 , etc. F or n = 1 , 2 , 3 it is kno wn ([2]) that cat(Sp( n )) = 1 , 3 , 5. Also, for n = 1 , 2 , 3 , 4 it is kno wn ([5]) th at cat(Sp in(2 n + 1)) = 1 , 3 , 5 , 8. Based on this small set of data, w e conjecture that cat(Sp( n )) = cat(Sp in(2 n + 1)) and that the inequalit y ab o ve is in fact an equalit y . W e remark that the b est kno wn lo wer b ound is cat(Sp( n )) ≥ n + 2 for n ≥ 3 ([2],[6]). 1.4. A cknow le dgment. The auth ors thank John Oprea who in tro duced u s to the p roblem during his visit to Ba ylor Univ ersit y in No ve mb er 2008. The authors also thank th e r eferee for p oin ting out an error in an earlier v ersion of the pap er. 2. Notation 2.1. Let G b e a simple, simply connected, compact Lie group with Lie algebra g . Let T b e a maximal torus of G w ith Lie algebra t . Th en h = t C is a Cartan subalgebra of g C with h R = i t . W rite ∆ = ∆( g C , h ) for the set of ro ots and choose a positive system ∆ + with corresp ondin g set of simple ro ots Π = { α 1 , . . . , α n } . With resp ect to this system, write α 0 for the highest root. F or the classical Lie groups and w ith resp ect to standard notation, Π and α 0 can b e tak en as in the follo wing table: G Π α 0 SU( n + 1) { α i = ε i − ε i +1 | 1 ≤ i ≤ n } ε 1 − ε n +1 Sp( n ) { α i = ε i − ε i +1 | 1 ≤ i ≤ n − 1 } S { α n = 2 ε n } 2 ε 1 Spin(2 n + 1) { α i = ε i − ε i +1 | 1 ≤ i ≤ n − 1 } S { α n = ε n } ε 1 + ε 2 Spin(2 n ) { α i = ε i − ε i +1 | 1 ≤ i ≤ n − 1 } S { α n = ε n − 1 + ε n } ε 1 + ε 2 2.2. W rite R ∨ for the coro ot lattice in h (whic h is the same as the dual to the w eigh t lattice in h ∗ ) s o that R ∨ = span Z { h α | α ∈ ∆ } . Here h α = 2 u α /B ( u α , u α ) ∈ h R where B ( · , · ) is th e K illing form and u α ∈ h R is un iquely determined b y the equation α ( H ) = B ( H , u α ) for all H ∈ h R . Sin ce G is simp ly connected, THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 3 it f ollo ws that k er (exp | t ) = 2 π i R ∨ . 2.3. The conn ected comp onent s of { t ∈ t | α ( t ) / ∈ 2 π i Z for α ∈ ∆ } are called alc oves . W rite W = W ( G, t ) for the W eyl group of G with resp ect to t view ed as acting on t (and extend ed to h as n eeded). The affine W eyl gr oup , c W , is the group generated by the transf ormations of t of the form t 7→ w t + z for w ∈ W and z ∈ k er (exp | t ). It ac ts acts simply transitiv ely on th e set of alco v es. The f undamental alc ove , A 0 , is the alco v e giv en by A 0 = { t = i H ∈ t | 0 < α ( H ) < 2 π for α ∈ ∆ + } = { t = i H ∈ t | α 0 ( H ) < 2 π and 0 < α j ( H ) f or 1 ≤ j ≤ n } . The closure of the fundamen tal alco v e, A 0 , is a fundamenta l domain for the c W -action (cf. [4, Th m. 4.8]). F or G = S p(2), th e ro ots and the fun dament al alc ov e are shown in Fig. 1. 3. Cells 3.1. Define v 0 = 0 ∈ t and f or 1 ≤ k ≤ n , define v k ∈ t by the equ ations α j ( v k ) =    2 π i if j = 0 0 if 1 ≤ j ≤ n and j 6 = k . Then { v 0 , . . . , v n } is the s et of v ertices of th e n -simp lex A 0 . Notice th at if we write α 0 = P n j =1 m j α j with m j ∈ N , we get 2 π i = α 0 ( v k ) = P n j =1 m j α j ( v k ) = m k α k ( v k ). T h erefore, α k ( v k ) = 2 π i m k for 1 ≤ k ≤ n. (F or classical G , the m k ∈ { 1 , 2 } ; ho we ver, for exceptional G , the m k can b e as large as 6.) 3.2. Define F 0 = { t = i H ∈ t | α 0 ( t ) = 2 π i and 0 ≤ α j ( H ) f or 1 ≤ j ≤ n } and f or 1 ≤ k ≤ n , F k = { t = i H ∈ t | α 0 ( H ) ≤ 2 π , 0 ≤ α j ( H ) f or 1 ≤ j ≤ n with j 6 = k , and 0 = α k ( t ) } 4 MARKUS HUNZIKER AND MARK R. SEP ANSKI α 0 α 2 α 1 Figure 1. Ro ots and alco v es for Sp(2) Then { F 0 , . . . , F n } is the set of faces of A 0 . F or 0 ≤ k ≤ n , w e will call F k the fac e opp osite to v k . In the follo wing, w e will write r k ∈ c W for the r efl ection across F k . Explicitly , r 0 ( t ) = t − ( α 0 ( t ) − 2 π i ) h α 0 and r k ( t ) = t − α k ( t ) h α k for 1 ≤ k ≤ n . 3.3. F or 0 ≤ k ≤ n , let c W k b e the stabilizer of v k , c W k = { w ∈ c W | w ( v k ) = v k } . Lemma 1. F or 0 ≤ k ≤ n , the gr oup c W k is gener ate d by { r j | 0 ≤ j ≤ n and j 6 = k } and { alc oves A such that v k ∈ A } = { w ( A 0 ) | w ∈ c W k } . Pr o of. F or the firs t statemen t, recall that it is well known (cf. [4, Ch. 4]) that the stabilizer of an y p oin t in A 0 is generated by the s et of r eflections across the alco v e faces that co ntain the p oin t. In p articular, v k lies on ev ery f ace except F k and the result follo ws. F or the second statemen t, obser ve that an y alco v e A can b e uniquely written as A = w ( A 0 ) for some w ∈ c W . Since th e vertic es of w ( A 0 ) are { w ( v j ) | 0 ≤ j ≤ n } , it follo ws that v k ∈ A if and only if v k = w ( v j ) f or some j , 0 ≤ j ≤ n . Since A 0 is a fu ndamenta l domain for the action of c W , v k = w ( v j ) if and only if k = j if and only if w ∈ c W k as d esired.  3.4. F or 0 ≤ k ≤ n , defin e C k = [ w ∈ c W k w  A 0 \ F k  . F or G = Sp(2), the cells are sho wn in Fig 2. By Lemma 1 and construction, th e follo wing result is immediate. Prop osition 2. (a) C k is an op en nei ghb orh o o d of v k that is c ontr actible to v k via a str a ig ht line c ontr action. THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 5 (b) Each alc ove wal l having nonempty interse ction with C k c onta ins v k . (c) Supp ose u 1 , u 2 ∈ C k satisfy u 2 = w ( u 1 ) for some w ∈ c W . Then v k = w ( v k ) . (d) A 0 ⊆ S n k =0 C k .  Figure 2. The cells C 0 , C 1 , and C 2 for S p(2) 4. A Co ver of G 4.1. F or 0 ≤ k ≤ n , defin e U k = { c g (exp t ) | g ∈ G , t ∈ C k } and O k = { c g (exp v k ) | g ∈ G } , where c g ( x ) = gxg − 1 for g , x ∈ G . Theorem 3. (a) { U k | 0 ≤ k ≤ n } is an op en c over of G . (b) O k is a deformation r etr act of U k . Pr o of. Sin ce exp( C k ) is op en in T and since conjugation tak es the exp onential of the closure of an alc ov e on to G , part (a) is automatic. F or part (b), we claim the deformation retract is giv en by R k : U k × I → U k where I = [0 , 1] and R k ( c g (exp t ) , s ) = c g (exp ((1 − s ) t + s v k )) . It remains to s ee that R k is actually w ell defined. Supp ose c g 1 (exp t 1 ) = c g 2 (exp t 2 ) for g j ∈ G and t j ∈ C k . W riting c g − 1 2 g 1 (exp t 1 ) = exp t 2 , there exists h ∈ Z G (exp t 2 ) 0 so that e w = hg − 1 2 g 1 ∈ N G ( T ) (cf. [8, Section 6.4].) Let Σ t 2 = { α ∈ ∆ | α ( t 2 ) ∈ 2 π i Z } , i.e., the set of α for which t 2 lies on an α -alco v e wall. 6 MARKUS HUNZIKER AND MARK R. SEP ANSKI Then Z G (exp t 2 ) 0 is the exp onen tial of the direct sum of t and all su (2)-triples corre- sp ond ing to ro ots in Σ t 2 . Since v k also lies on all suc h α -alco v e wall s, it follo ws that h ∈ Z G (exp ((1 − s ) t + s v k )) 0 . Setting w = Ad e w ∈ W , w e ha v e c e w (exp t 1 ) = exp t 2 . Th us exp( w t 1 ) = exp( t 2 ) so that t 2 = w t 1 + z f or some z ∈ ker ( exp | t ). By Prop osition 2 , it follo ws that v k = wv k + z . Then c g 1 (exp ((1 − s ) t 1 + sv k )) = c g 2 h − 1 e w (exp ((1 − s ) t 1 + sv k )) = c g 2 h − 1 (exp ((1 − s ) wt 1 + sw v k )) = c g 2 h − 1 (exp ((1 − s ) ( t 2 − z ) + s ( v k − z ))) = c g 2 h − 1 (exp ((1 − s ) t 2 + sv k − z )) = c g 2 (exp ((1 − s ) t 2 + sv k )) and we are finished.  4.2. The r esults of the pr evious su b section giv e immediately the follo win g m ain result. Theorem 4. cat( G ) + 1 ≤ n X k =0 (cat G ( O k ) + 1) .  5. The O rbits O k W e presen t some remarks and explicit realizations for th e O k in the classica l cases. 5.1. G = SU( n + 1). T rivial calculations sho w that v k = 2 π i n + 1 ( k z }| { n + 1 − k , . . . , n + 1 − k , − k , . . . , − k ) for 0 ≤ k ≤ n . Therefore exp v k = e − 2 π i k n +1 Id. In p articular, O k = { e − 2 π i k n +1 Id } and so cat ( O k ) = 0 for all 0 ≤ k ≤ n . Thus, Th eorem 4 implies cat(SU( n + 1)) ≤ n , i.e., w e reco v er Singhof ’s result [9]. 5.2. G = Sp( n ). Let H denote th e division algebra of quaternions q = a + b i + c j + d k , a, b, c, d ∈ R . View H n as a r igh t v ector space and iden tify th e set of quaternionic matrices, M n ( H ), with the set of H -linear endomorphisms of H n via standard matrix multiplic ation on THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 7 the left. W rite ν : M n ( H ) → R for the reduced n orm. In particular, if ϕ : M n ( H ) → M 2 n ( C ) is the C -linear injecti ve h omomorphism give n by ϕ ( A + j B ) = A B − ¯ B ¯ A ! for A, B ∈ M n ( C ), then ν = det ◦ ϕ . W e then realize GL( n, H ) = { g ∈ M n ( H ) | ν ( g ) 6 = 0 } , SL( n, H ) = { g ∈ M n ( H ) | ν ( g ) = 1 } , and G = Sp( n ) = { g ∈ SL( n, H ) | g g ∗ = I n } , where g ∗ denotes the quaternionic conju gate tran s p ose of g . W e also fix th e m aximal torus T = { diag( e i θ 1 , . . . , e i θ n ) | θ j ∈ R } . With this set-up, it is s traigh tforwa rd to c hec k that v k = i π diag( k z }| { 1 , . . . , 1 , 0 , . . . , 0) for 0 ≤ k ≤ n . Th er efore exp v k = − I k I n − k ! . In p articular, O 0 = { Id } and O n = {− I d } so that cat ( O 0 ) = cat ( O n ) = 0. The other O k require more work, though they are easy to ident ify . F o r this w e reali ze the q u aternionic Gr assm annian of k -planes in H n , Gr k ( H n ), by { x ∈ M n × k ( H ) | rk( x ) = k } equipp ed with the equiv alence relation x ∼ xh where x ∈ M n × k ( H n ) and h ∈ GL( k , H ). The f ollo win g result is immediate. Lemma 5. L et 1 ≤ k ≤ n − 1 and set d k = min { k, n − k } . Then ther e is a diffe omorph ism τ k : O k → Gr d k ( H n ) , O k ∼ = Sp( n ) / (Sp( k ) × Sp( n − k )) ∼ = Gr d k ( H n ) , given by τ k ( c g (exp v k )) = g I k 0 ( n − k ) × k ! when d k = k and by τ k ( c g (exp v k )) = g 0 k × ( n − k ) I n − k ! when d k = n − k .  Conjecture 1. cat Sp( n ) ( O k ) = d k . 8 MARKUS HUNZIKER AND MARK R. SEP ANSKI As w e observ ed already in the in tro duction, if the conjecture is true, then Theorem 4 quic kly sho ws that cat (Sp( n )) ≤  ( n + 2) 2 4  − 1 . In terms of trying to sho w that cat Sp( n ) ( O ) k ≤ d k , there is an ob vious c hoice of a co ve r of O k . F or this, we in tro du ce the f ollo win g notation. F or the sak e of clarit y , w e assume we are in the case of d k = k , i.e., 1 ≤ k ≤ n/ 2. F or 1 ≤ j ≤ k + 1, wr ite x ∈ Gr k − 1 ( H n − 1 ) as x = x j, 1 x j, 2 ! with x j, 1 ∈ M ( j − 1) × ( k − 1) ( H ) and x j, 2 ∈ M ( n − j ) × ( k − 1) ( H ). Let X j,k ∼ = Gr k − 1 ( H n − 1 ) ⊆ Gr k ( H n ) b e giv en by {    0 ( j − 1) × 1 x j, 1 1 0 1 × ( k − 1) 0 ( k − j ) × 1 x j, 2    | x ∈ Gr k − 1 ( H n − 1 ) } . W rite y ∈ Gr k ( H n − 1 ) as y = y j, 1 y j, 2 ! with y j, 1 ∈ M ( j − 1) × k ( H ) and y j, 2 ∈ M ( n − j ) × k ( H ). Let Y j,k ∼ = Gr k ( H n − 1 ) ⊆ Gr k ( H n ) b e giv en b y {    y j, 1 0 1 × k y j, 2    | y ∈ Gr k ( H n − 1 ) } . Prop osition 6. (a) { Gr k ( H n ) \ X j,k | 1 ≤ j ≤ k + 1 } is an op en c over of Gr k ( H n ) . (b) Y j,k is a deformation r etr act of Gr k ( H n ) \ X j,k . (c) Written i n ( j − 1) × 1 × ( n − j ) blo ck form, τ − 1 k ( Y j,k ) is {    A B 1 C D    | A B C D ! ∈ S p( n − 1) and c onjugate to exp v k − 1 ,n − 1 } wher e v k ,n = i diag( k z }| { π , . . . , π , n − k z }| { 0 , . . . , 0) . THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 9 Pr o of. F or part (a), simply obs er ve th at a k -plane in X 1 ,k ∩ · · · ∩ X k +1 ,k w ould hav e to con tain k + 1 indep endent v ectors whic h is imp ossib le. F or part (b), observe that Gr k ( H n ) \ X j,k is the of the s et of    x ( j − 1) × k y 1 × k z ( n − j ) × k    ∈ Gr k ( H n ) s o th at x ( j − 1) × k z ( n − j ) × k ! ∈ Gr k ( H n − 1 ) . Therefore, the retraction R : Gr k ( H n ) \ X j,k × I → X j,k giv en b y R (    x ( j − 1) × k y 1 × k z ( n − j ) × k    , s ) =    x ( j − 1) × k (1 − s ) y 1 × k z ( n − j ) × k    do es the trick. F or part (c), obser ve that τ − 1 k ( Y j,k ) can b e wr itten in ( j − 1) × 1 × ( n − k ) blo c k form as { g =    α β γ 0 δ ζ η ι κ    ∈ G } . Making note that g g ∗ = I , p art (c) follo ws immediate ly b y explicit matrix multiplicat ion using ( j − 1) × 1 × ( k − j ) × ( n − j ) b lo c k form when j ≤ k and by u sing k × 1 × ( n − k − 1) blo c k form w hen j = k + 1.  Prop osition 7. If the sets τ − 1 k ( Y j,k ) ar e c ontr actible in SL( n, H ) , then cat Sp( n ) ( O k ) ≤ k . Pr o of. Let F 1 : τ − 1 k ( Y j,k ) × I → SL ( n , H ) b e a con traction that tak es τ − 1 k ( Y j,k ) to a p oint. Using the Cartan decomp osition, th ere is a diffeomorph ism SL( n, H ) ∼ = G × p wh ere p is the the − 1 eigenspace of the Cartan inv olution corresp ondin g to sp ( n ), i.e., the in v olution giv en b y θ ( x ) = − x ∗ . F or g ∈ SL( n, H ), u niquely write g = κ ( g ) exp ( ρ ( g )) with κ ( g ) ∈ G and ρ ( g ) ∈ p . Finally , define F 2 : τ − 1 k ( Y j,k ) × I → G b y F 2 ( g , s ) = κ ( F 1 ( g , s )) . By construction, F 2 con tracts τ − 1 k ( Y j,k ) to a p oint . T hus, if the sets τ − 1 k ( Y j,k ) are con tractible in S L( n, H ) then they are also con tractible in G = Sp( n ). The prop osition then follo ws from Prop osition 6.  A t the pr esen t time, we do n ot kn o w whether τ − 1 k ( Y j,k ) is con tractible in S L ( n, H ). It is w orth noting that a similar resu lt can b e obtained by sho w in g that τ − 1 k ( Y j,k ) is contract ib le in S p(2 n, C ). Th is too is un kno wn . 10 MARKUS HUNZIKER AND MARK R. SEP ANSKI 5.3. G = Spin(2 n + 1). W rite the tensor algebra o v er R m as T m ( R ). Th en the Clifford algebra is C m ( R ) = T m ( R ) / I where I is the ideal of T m ( R ) generated by { ( x ⊗ x + k x k 2 ) | x ∈ R m } . By wa y of n otation for Clifford multiplicati on, w rite x 1 x 2 · · · x k for the elemen t x 1 ⊗ x 2 ⊗ · · · ⊗ x k + I ∈ C m ( R ) where x 1 , x 2 , . . . , x m ∈ R m . W rite C + m ( R ) f or the subalgebra of C m ( R ) spanned b y all pr o ducts of an ev en num b er of elemen ts of R m . Co n jugation, an an ti-in volutio n on C m ( R ), is defined by ( x 1 x 2 · · · x k ) ∗ = ( − 1) k x k · · · x 2 x 1 for x i ∈ R m . Then Spin( m ) = { g ∈ C + m ( R ) | gg ∗ = 1 and gxg ∗ ∈ R m for all x ∈ R m } . In fact, it is th e case that Sp in( m ) = { x 1 x 2 · · · x 2 k | x i ∈ S m − 1 for 2 ≤ 2 k ≤ 2 m } . If we write ( A g ) x = g xg ∗ when g ∈ Sp in( m ) and x ∈ R m , then A giv es the d ouble cov er of SO( m ): { 1 } → {± 1 } → Spin( m ) A → SO( m ) → { I m } . A maximal torus T 0 for S O (2 n + 1) is giv en b y T 0 = {             cos θ 1 sin θ 1 − sin θ 1 cos θ 1 . . . cos θ n sin θ n − sin θ n cos θ n 1             | θ i ∈ R } with Lie algebra t 0 = {             0 θ 1 − θ 1 0 . . . 0 θ n − θ n 0 0             | θ i ∈ R } . W e write exp SO(2 n +1) for the exp onen tial map from t 0 on to T 0 and condense notation b y writing E k for the elemen t of t giv en b y E k = blo c kdiag  k z }| { 0 0 0 0 ! , . . . , 0 0 0 0 ! , 0 1 − 1 0 ! , 0 0 0 0 ! , . . . , 0 0 0 0 ! , 0  . THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 11 W riting e k for the k th standard basis v ector in R n , observ e that A (cos θ − sin θ e 2 k − 1 e k ) acts by the rotati on cos 2 θ sin 2 θ − sin 2 θ cos 2 θ ! in the e 2 k − 1 e k plane. It follo ws that T = { (cos θ 1 − sin θ 1 e 1 e 2 ) · · · (cos θ n − sin θ n e 2 n − 1 e 2 n ) | θ k ∈ R } is a maximal toru s of Spin(2 n + 1). If we identify t w ith the Lie algebra of T and write exp for the exp onen tial map of S p in(2 n + 1) taking t onto T , then exp SO( n ) = A ◦ exp. It follo ws that exp ( θ E k ) = (cos( θ / 2) − sin( θ / 2) e 2 k − 1 e 2 k ) . Using the definitions, it is straightfo r ward to c hec k that v 0 = 0 v 1 = 2 π E 1 v k = π k X j =1 E j for 2 ≤ k ≤ n . Therefore exp v 0 = 1, exp v 1 = − 1, and exp v k = ( − 1) k Q k j =1 e 2 j − 1 e j . Of course, O 0 = { 1 } and O 1 = {− 1 } so cat( O 0 ) = cat( O 1 ) = 0. The other orbits are easy to describ e, though calculati n g cat G ( O k ) is not easy . Prop osition 8. F or 2 ≤ k ≤ n , O k ∼ = Spin(2 k ) / Spin (2 k ) Sp in(2 n + 1 − 2 k ) ∼ = SO(2 n + 1) / ( S O(2 k ) × SO(2 n + 1 − 2 k )) ∼ = ] Gr 2 k ( R 2 n +1 ) , the Gr assm annian of oriente d 2 k -planes i n R 2 n +1 . Pr o of. Sin ce A (exp v k ) = − I 2 k I n − 2 k ! , A ( O k ) ∼ = SO(2 n + 1) /S (O(2 k ) × O(2 n + 1 − 2 k )) ∼ = Gr 2 k ( R 2 n +1 ) , the Gr assmannian of 2 k -planes in R 2 n +1 . Moreo ve r , A : O k → A ( O k ) is a doub le cov er. T o see this, observe that there is a W eyl group (isomorphic to S n ⋉ Z n 2 ) elemen t taking v k to − π E 1 + π P k j =2 E j whic h exp onentiat es to − exp v k . T o p ro ve the p rop osition, fir st observe that the stabilizer of exp v k under conju gation m ust b e con tained in S = A − 1 ( S (O(2 k ) × O(2 n + 1 − 2 k )) ) = S (Pin(2 k ) Pin (2 n + 1 − 2 k )) . Since Pin(2 k ) ∩ Pin(2 n + 1 − 2 k ) ⊆ R , it follo ws that the connected comp onent of the iden tit y of S is S 0 = Spin(2 k ) Sp in( n − 2 k ) ∼ = Spin(2 k ) × Spin( n − 2 k ) / {± (1 , 1) } and the 12 MARKUS HUNZIKER AND MARK R. SEP ANSKI other comp onent is diffeomorphic to Pin (2 k ) 1 × Pin(2 n + 1 − 2 k ) 1 where Pin( j ) 1 is the non-iden tity component of Pin( j ). Recalling that th e cente r of Spin(2 k ) is {± 1 , ± exp v k } , it f ollo ws th at S 0 is cont ained in the stabilize r of exp v k . Ho wev er, Pin(2 k ) 1 an ticomm utes with exp v k while Pin(2 n + 1 − 2 k ) 1 comm utes. Th erefore, the stabiliz er of exp v k is S 0 . Finally , since S 0 = A − 1 (SO(2 k ) × S O( n − 2 k )), the p ro of is complete.  The relativ e cat ca lculation of O k in S pin(2 n + 1) is not kn o wn. 5.4. G = Spin(2 n ). A maximal torus T 0 for S O(2 n ) is given by T 0 = {          cos θ 1 sin θ 1 − sin θ 1 cos θ 1 . . . cos θ n sin θ n − sin θ n cos θ n          | θ i ∈ R } with Lie algebra t = {          0 θ 1 − θ 1 0 . . . 0 θ n − θ n 0          | θ i ∈ R } . As b efore, write E k = blo c kdiag       k z }| { 0 0 0 0 ! , . . . , 0 0 0 0 ! , 0 1 − 1 0 ! , 0 0 0 0 ! , . . . , 0 0 0 0 !       . F rom th e defin itions, it is straightforw ard to c hec k that v 0 = 0 v 1 = 2 π E 1 v k = π k X j =1 E j v n − 1 = π n − 1 X j =1 E j − π E n THE L-S CA TEGOR Y O F SIMPL Y CON NECTED CO MP A CT LIE GR OUPS 13 for 2 ≤ k ≤ n , k 6 = n − 1. Th erefore exp v 0 = 1, exp v 1 = − 1, exp v k = ( − 1) k Q k j =1 e 2 j − 1 e j , and exp v n − 1 = ( − 1) n − 1 Q n j =1 e 2 j − 1 e j .Of course, O 0 = { 1 } and O 1 = {− 1 } so cat ( O 0 ) = cat( O 1 ) = 0. As in Prop ostion 8, the remaining conjugacy classes are O k ∼ = Spin(2 k ) / Spin 2 k ( R ) Spin(2 n − 2 k ) ∼ = SO(2 n ) / S O(2 k ) × SO(2 n − 2 k ) ∼ = ] Gr 2 k ( R 2 n ), the Grassmannian of oriented 2 k -p lanes in R 2 n . Again, the relativ e category in Spin(2 n ) is not kno wn. Referen ces [1] O. Cornea, G. Lu p ton, J. Oprea, and D. T anr´ e: Lusternik-Schnirelmann category . Mathematical Su rveys and Monographs, 103. American Mathematical S o ciety , Pro v id ence, RI, 2003. xviii+330 pp. [2] L. F ernandez-S uarez, A. Gomez-T ato, J. S trom, and D. T anr´ e: The Lusternik-Schnir el mann c ate gory of Sp(3), T rans. Amer. Math. So c. 132 (2004), 587595. [3] T. Ganea: Some pr oblems on numeric al homotopy i nvariants , in: S ymp osium on Algebraic T op ology , in: Lecture Notes in Math., vol. 249, Springer, Berlin, 1971, pp. 1322 . [4] J. E. Humphreys: Reflection groups and Coxeter groups. 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Z. 151 (1976) 143–148. Dep ar tment of Ma thema tics, Ba ylor U niversity, W a co, Texas E-mail addr ess : { Mark us Hunziker, Mark Sepanski } @bay lor.edu