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A Note on Kottwitz's Invariante(G)
208, 372᎐377 Ž1998. JA987506
JOURNAL OF ALGEBRA ARTICLE NO.
A Note on Kottwitz’s Invariant eŽ G . Wee Teck Gan Mathematics Department, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138 Communicated by J. Saxl Received December 13, 1997
Kottwitz has defined an invariant eŽ G . for any reductive group G. In this note, we give an interpretation of eŽ G . in terms of the Killing forms of G and its quasi-split inner form. 䊚 1998 Academic Press
1. THE INVARIANTS dŽ G . AND eŽ G . Let G be a connected reductive group over a field k with char Ž k . / 2, and let Gs be its split form. Hence, G defines an element aG in H 1 Ž k, Autk s Ž Gs ... Let Gsad be the quotient of Gs by its center. Also, let ⌽ be the based root datum of Gs . Then we have the following exact sequence of Galois modules: 1 ª Gsad ª Autk s Ž Gs . ª Aut Ž ⌽ . ª 1. We also have a determinant map det: Aut Ž ⌽ . ª ² "1:. From the composite, d: H 1 Ž k, Autk s Ž Gs . . ª H 1 Ž k, Autk s Ž ⌽ . . ª H 1 Ž k, ² "1: . , we get an elementary cohomological invariant of G, which is given by d Ž G . s d Ž aG . g H 1 Ž k, ² "1: . s H 1 Ž k , 2 . , since char Ž k . / 2. Here, the groups or pointed sets H *Ž k, y. refer to those of Galois cohomology, or equivalently, to those with respect to the ´etale topology on SpecŽ k .. 372 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
373
On the other hand, Kottwitz wKx has defined an invariant eŽ G . g H 2 Ž k, 2 . as follows. Let G⬘ be the quasi-split inner form of G. So G gives X X X .-orbit in H 1 Ž k, Gad .. Here again, Gad a well-defined Autk Ž Gad is the quotient of G⬘ by its center. From the exact sequence, X X 1 ª Z ª Gsc ª Gad ª 1,
of sheaves in the flat topology on SpecŽ k ., we get a map X X ␦ : H 1 Ž k, Gad . ª H f1 Ž k , Gad . ª H f2 Ž k, Z . ,
where H fU Ž k, y. refers to flat cohomology. Let ŽT, B . be a pair consisting of a maximal torus and a Borel subgroup X of Gsc . Let be the character of T given by half the sum of the positive roots Žpositive with respect to B ., and let be the restriction of to Z. Then Kottwitz showed that is independent of the choice of ŽT, B ., is X .. Moreover, defined over k, and is stable under the action of Autk Ž Gsc 2 since is the identity on Z, we have : Z ª 2 . Hence the composite, X ( ␦ : H 1 Ž k, Gad . ª H f2 Ž k, Z . ª H f2 Ž k, 2 . ( H 2 Ž k, 2 . , X . gives an invariant eŽ G . s ( ␦ Ž cG ., where cG is any element of H 1 Ž k, Gad in the orbit determined by G. As discussed in wKx, the need to use flat cohomology arises because char Ž k . may divide the order of Z. Over a local field, the invariant eŽ G . plays a role in the study of the trace formula and orbital integrals of G. Now, let W Ž k . be the Witt ring of nondegenerate quadratic forms over k, and let I be the ideal of even rank forms wSx. Then IrI 2 ( H 1 Ž k, 2 . and I 2rI 3 ( H 2 Ž k, 2 .. Hence the invariants dŽ G . and eŽ G . can be regarded as elements in IrI 2 and I 2rI 3, respectively. It is thus natural to ask whether they can be interpreted in terms of quadratic forms.
2. KILLING FORM Now assume that G is semisimple and that the Killing form BG of G is nondegenerate on the Lie algebra ᒄ of G. This latter assumption holds, for example, when char Ž k . s 0, or char Ž k . is greater than the Coxeter numbers of each of the simple factors of G. Let O denote the orthogonal group of BG s and SO its connected component Žof index 2.. We have a natural morphism Autk s Ž Gs . ª O such that its restriction to Gsad is simply the adjoint representation Ad: Gsad ª SO. We thus have the following
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commutative diagram of Galois modules with exact rows: 1
ª
1
ª
ª
Gsad x SO
ª
Autk s Gs x O
ª
Aut ⌽ x ² "1:
ª
ª
1
ª
1
From this, we get the following commutative diagram: H 1 Ž k , Autk sGs . Ad
ª
x
H 1 Ž k, Aut ⌽ . x
H 1 Ž k, O .
det
ª
det
H 1 Ž k, ² " 1: .
This implies that dŽ G . s det ( AdŽ aG .. One checks that AdŽ aG . g H 1 Ž k, O . is simply the class corresponding to the Killing form BG of G, and that det ( Ad Ž aG . s det Ž BG . rdet Ž BG s . . Hence dŽ G . corresponds to the element Ž ᒄ [ ᒄ s , BG [ BG s . g I. Remarks. The assumption that G is semisimple is for convenience. If G is reductive, say G s Z ⭈ G 0 , with Z central and G 0 the derived group of G, then we have d Ž G . rd Ž Z . s det Ž BG 0 . rdet Ž BŽG 0 . s . . Now, for eŽ G ., we shall use the following commutative diagram with exact rows of sheaves in the flat topology: X X ª Gad 1 ª Z ª Gsc ª1
x ⬘ x x Ad 1 ª 2 ªSpin ª SO ª1 Here, SO denotes the special orthogonal group for the Killing form BGXa d . This gives X H f1 Ž k, Gad .
x
␦
ª
x
Ad
H f1 Ž k, SO .
H f2 Ž k, Z .
␦
ª
⬘
H f2 Ž k, 2 .
We define: e⬘ Ž G . s ␦ ( Ad Ž cG . g H f2 Ž k, 2 . ( H 2 Ž k, 2 . ,
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
375
X . representing G, or rather its image in where cG is any class in H 1 Ž k, Gad X 1Ž H f k, Gad .. Note that the definition of e⬘ uses only the Spin exact sequence. Hence, since char Ž k . / 2 by assumption, e⬘ can be defined purely in terms of Galois cohomology. Moreover, the above commutative diagram shows that e⬘Ž G . s ⬘( ␦ Ž cG .. Of course, we have to show that e⬘Ž G . is well defined, i.e., independent of the choice of cG . To see this, note that if bG is another class in X . representing G, then AdŽ bG . and AdŽ cG . are two classes in H 1 Ž k, Gad 1Ž H k, SO ., which have the same image in H 1 Ž k, O ., since they both represent the Killing form BG . From the commutative diagram
1 ª 2 ªSpin ªSO ª1 x id x x 1 ª 2 ª Pin ª O ª1 we have H 1 Ž k , SO . x
ª
H 2 Ž k, 2 . x id
H 1 Ž k, O .
ª
H 2 Ž k, 2 .
Hence the images of AdŽ bG . and AdŽ cG . in H 2 Ž k, 2 . are the same, i.e., eX Ž G . is well defined. Now for any reductive group G with center C, we set e⬘Ž G . [ e⬘Ž GrC .. The following properties of e⬘Ž G . are easy to check: PROPOSITION 1. Ži. e⬘Ž G . s w Ž BG .rw Ž BG ⬘ ., where w Ž Q . is the Hasse᎐Witt in¨ ariant of the quadratic form Q Ž see w S x for the definition of w, and wSex for this equality .. Žii. e⬘Ž G1 = G 2 . s e⬘Ž G1 . e⬘Ž G 2 ., using multiplicati¨ e notation in H 2 Ž k, 2 .. Žiii. e⬘Ž G . s e⬘Ž GrZ ., where Z is central in G. Živ. e⬘Ž G mk E . s ResŽ e⬘Ž G .., where E is a field extension of k, and Res: H 2 Ž k, 2 . ª H 2 Ž E, 2 . is the usual restriction map. Žv. e⬘Ž ResEr k GE . s Cor Ž e⬘Ž GE .., where E is a finite extension of k and Cor: H 2 Ž E, 2 . ª H 2 Ž k, 2 . is the usual corestriction map. We want to show that eŽ G . s e⬘Ž G .. It suffices to show that s ⬘. By Proposition 1, we only have to do this for G absolutely quasi-simple. PROPOSITION 2.
s ⬘.
Proof. This is just a question about coweight and coroot lattices. Indeed, the adjoint representation ad: ᒄ⬘ ª ᒐ ᒌ induces a map from the coroot lattice nr of ᒄ⬘ to the coroot lattice ⌳ r of ᒐ ᒌ. It also maps the
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coweight lattice nw of ᒄ⬘ into a lattice ⌳ that contains ⌳ r as a sublattice of index 2. ⌳ is simply the lattice corresponding to the group SO. Note that if dimŽ ᒄ⬘. is odd, then ⌳ is simply the coweight lattice. Otherwise, it is a lattice of index 2 in the coweight lattice. Hence, we get an induced map: nw rnr ª ⌳r⌳ r . This map of lattices induces the map ⬘ of finite group schemes, and so we denote it by ⬘ as well. Indeed, if nUw Žrespectively nUr . denotes the root Žrespectively weight. lattice, then the finite group scheme Z is the Cartier dual of the constant group scheme nUr rnUw . To compute this map of lattices, it suffices to assume that ᒄ⬘ is defined over ⺓, so that we can regard the lattices above as sitting in ᒄ⬘ and ᒐ ᒌ. So let H␣ , X␣ , Y␣ 4 be a Chevalley basis of ᒄ⬘. Using the isomorphism ᒐ ᒌ ( n2 ᒄ⬘ described in wF-H, p. 303x, we see that for any t g ᒑ, the Lie algebra of the maximal torus of G⬘, we have ad: t ¬
Ý
␣g⌽ q
␣ Ž t.
1 2
X␣ n Y␣ g ᒑ Ž ᒐ ᒌ . .
Let t␣ s 12 X␣ n Y␣ . Then the t␣ ’s lie in ⌳ and can be completed to a basis of ⌳. Furthermore, the intersection of the ⺪-span of the t␣ ’s with ⌳ r is generated by t␣ " t : ␣ ,  g ⌽q4 . Notice that if t g nw , then ␣ Ž t . g ⺪ for all ␣ , and so adŽ t . g ⌳. We can check if an element of nw lands in ⌳ r as follows. There is an augmentation map:
: ⺪ w t␣ x ª 12 ⺪ cs Ý c␣ t␣ ¬
1 2
Ý c␣ .
Then c lies in ⌳ r if and only if Ž c . g ⺪, i.e.,
: ⺪ w t␣ x r⌳ r l ⺪ w t␣ x ( 12 ⺪r⺪. Hence adŽ t . lies in ⌳ r if and only if 12 Ý ␣ g ⌽ q ␣ Ž t . g ⺪, i.e., if and only if Ž t . g ⺪. Hence ( ⬘ s < Z , as required. COROLLARY 3.
eŽ G . s e⬘Ž G ..
EXAMPLE. Suppose k s ⺢. Then we have the Cartan decomposition of ᒄ and ᒄ⬘: ᒄsᒈ[ᒍ ᒄ⬘ s ᒈ⬘ [ ᒍ⬘.
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
377
The Killing form is positive definite Žrespectively, negative definite. on ᒍ and ᒍ⬘ Žrespectively, ᒈ and ᒈ⬘.. Let n s dimŽ ᒈ . and m s dimŽ ᒈ⬘.. Then we have w Ž BG . s Ž y1 .
1 2
n Ž ny1 .
w Ž BG⬘ . s Ž y1 .
1 2
m Ž my1 .
.
Since G and G⬘ are inner twists of each other, n and m must have the same parity. Hence, w Ž BG . rw Ž BG⬘ . s Ž y1 .
1 2
Ž nym .
s Ž y1 .
1 2
Ž di m GrKydi m G⬘rK ⬘ .
,
where K Žrespectively, K ⬘. is the maximal compact subgroup of G Žrespectively, G⬘.. Remarks. Ž1. This computation of eŽ G . for k s ⺢ seems to be simpler than that given in wKx. Ž2. It would be nice to show, using this definition, that if k is p-adic, then e⬘Ž G . is as computed in wKx. However, I am unable to prove this.
REFERENCES wF-Hx W. Fulton and J. Harris, Representation theory, Grad. Texts in Math. 129. wKx R. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 Ž1983., 289᎐297. wSx W. Scharlau, Quadratic and Hermitian forms, Grundlehren Math. Wiss. 270. wSex J. P. Serre, Cohomologie Galoisienne, Lecture Notes in Math. 5.
JOURNAL OF ALGEBRA ARTICLE NO.
A Note on Kottwitz’s Invariant eŽ G . Wee Teck Gan Mathematics Department, Har¨ ard Uni¨ ersity, Cambridge, Massachusetts 02138 Communicated by J. Saxl Received December 13, 1997
Kottwitz has defined an invariant eŽ G . for any reductive group G. In this note, we give an interpretation of eŽ G . in terms of the Killing forms of G and its quasi-split inner form. 䊚 1998 Academic Press
1. THE INVARIANTS dŽ G . AND eŽ G . Let G be a connected reductive group over a field k with char Ž k . / 2, and let Gs be its split form. Hence, G defines an element aG in H 1 Ž k, Autk s Ž Gs ... Let Gsad be the quotient of Gs by its center. Also, let ⌽ be the based root datum of Gs . Then we have the following exact sequence of Galois modules: 1 ª Gsad ª Autk s Ž Gs . ª Aut Ž ⌽ . ª 1. We also have a determinant map det: Aut Ž ⌽ . ª ² "1:. From the composite, d: H 1 Ž k, Autk s Ž Gs . . ª H 1 Ž k, Autk s Ž ⌽ . . ª H 1 Ž k, ² "1: . , we get an elementary cohomological invariant of G, which is given by d Ž G . s d Ž aG . g H 1 Ž k, ² "1: . s H 1 Ž k , 2 . , since char Ž k . / 2. Here, the groups or pointed sets H *Ž k, y. refer to those of Galois cohomology, or equivalently, to those with respect to the ´etale topology on SpecŽ k .. 372 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
373
On the other hand, Kottwitz wKx has defined an invariant eŽ G . g H 2 Ž k, 2 . as follows. Let G⬘ be the quasi-split inner form of G. So G gives X X X .-orbit in H 1 Ž k, Gad .. Here again, Gad a well-defined Autk Ž Gad is the quotient of G⬘ by its center. From the exact sequence, X X 1 ª Z ª Gsc ª Gad ª 1,
of sheaves in the flat topology on SpecŽ k ., we get a map X X ␦ : H 1 Ž k, Gad . ª H f1 Ž k , Gad . ª H f2 Ž k, Z . ,
where H fU Ž k, y. refers to flat cohomology. Let ŽT, B . be a pair consisting of a maximal torus and a Borel subgroup X of Gsc . Let be the character of T given by half the sum of the positive roots Žpositive with respect to B ., and let be the restriction of to Z. Then Kottwitz showed that is independent of the choice of ŽT, B ., is X .. Moreover, defined over k, and is stable under the action of Autk Ž Gsc 2 since is the identity on Z, we have : Z ª 2 . Hence the composite, X ( ␦ : H 1 Ž k, Gad . ª H f2 Ž k, Z . ª H f2 Ž k, 2 . ( H 2 Ž k, 2 . , X . gives an invariant eŽ G . s ( ␦ Ž cG ., where cG is any element of H 1 Ž k, Gad in the orbit determined by G. As discussed in wKx, the need to use flat cohomology arises because char Ž k . may divide the order of Z. Over a local field, the invariant eŽ G . plays a role in the study of the trace formula and orbital integrals of G. Now, let W Ž k . be the Witt ring of nondegenerate quadratic forms over k, and let I be the ideal of even rank forms wSx. Then IrI 2 ( H 1 Ž k, 2 . and I 2rI 3 ( H 2 Ž k, 2 .. Hence the invariants dŽ G . and eŽ G . can be regarded as elements in IrI 2 and I 2rI 3, respectively. It is thus natural to ask whether they can be interpreted in terms of quadratic forms.
2. KILLING FORM Now assume that G is semisimple and that the Killing form BG of G is nondegenerate on the Lie algebra ᒄ of G. This latter assumption holds, for example, when char Ž k . s 0, or char Ž k . is greater than the Coxeter numbers of each of the simple factors of G. Let O denote the orthogonal group of BG s and SO its connected component Žof index 2.. We have a natural morphism Autk s Ž Gs . ª O such that its restriction to Gsad is simply the adjoint representation Ad: Gsad ª SO. We thus have the following
374
WEE TECK GAN
commutative diagram of Galois modules with exact rows: 1
ª
1
ª
ª
Gsad x SO
ª
Autk s Gs x O
ª
Aut ⌽ x ² "1:
ª
ª
1
ª
1
From this, we get the following commutative diagram: H 1 Ž k , Autk sGs . Ad
ª
x
H 1 Ž k, Aut ⌽ . x
H 1 Ž k, O .
det
ª
det
H 1 Ž k, ² " 1: .
This implies that dŽ G . s det ( AdŽ aG .. One checks that AdŽ aG . g H 1 Ž k, O . is simply the class corresponding to the Killing form BG of G, and that det ( Ad Ž aG . s det Ž BG . rdet Ž BG s . . Hence dŽ G . corresponds to the element Ž ᒄ [ ᒄ s , BG [ BG s . g I. Remarks. The assumption that G is semisimple is for convenience. If G is reductive, say G s Z ⭈ G 0 , with Z central and G 0 the derived group of G, then we have d Ž G . rd Ž Z . s det Ž BG 0 . rdet Ž BŽG 0 . s . . Now, for eŽ G ., we shall use the following commutative diagram with exact rows of sheaves in the flat topology: X X ª Gad 1 ª Z ª Gsc ª1
x ⬘ x x Ad 1 ª 2 ªSpin ª SO ª1 Here, SO denotes the special orthogonal group for the Killing form BGXa d . This gives X H f1 Ž k, Gad .
x
␦
ª
x
Ad
H f1 Ž k, SO .
H f2 Ž k, Z .
␦
ª
⬘
H f2 Ž k, 2 .
We define: e⬘ Ž G . s ␦ ( Ad Ž cG . g H f2 Ž k, 2 . ( H 2 Ž k, 2 . ,
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
375
X . representing G, or rather its image in where cG is any class in H 1 Ž k, Gad X 1Ž H f k, Gad .. Note that the definition of e⬘ uses only the Spin exact sequence. Hence, since char Ž k . / 2 by assumption, e⬘ can be defined purely in terms of Galois cohomology. Moreover, the above commutative diagram shows that e⬘Ž G . s ⬘( ␦ Ž cG .. Of course, we have to show that e⬘Ž G . is well defined, i.e., independent of the choice of cG . To see this, note that if bG is another class in X . representing G, then AdŽ bG . and AdŽ cG . are two classes in H 1 Ž k, Gad 1Ž H k, SO ., which have the same image in H 1 Ž k, O ., since they both represent the Killing form BG . From the commutative diagram
1 ª 2 ªSpin ªSO ª1 x id x x 1 ª 2 ª Pin ª O ª1 we have H 1 Ž k , SO . x
ª
H 2 Ž k, 2 . x id
H 1 Ž k, O .
ª
H 2 Ž k, 2 .
Hence the images of AdŽ bG . and AdŽ cG . in H 2 Ž k, 2 . are the same, i.e., eX Ž G . is well defined. Now for any reductive group G with center C, we set e⬘Ž G . [ e⬘Ž GrC .. The following properties of e⬘Ž G . are easy to check: PROPOSITION 1. Ži. e⬘Ž G . s w Ž BG .rw Ž BG ⬘ ., where w Ž Q . is the Hasse᎐Witt in¨ ariant of the quadratic form Q Ž see w S x for the definition of w, and wSex for this equality .. Žii. e⬘Ž G1 = G 2 . s e⬘Ž G1 . e⬘Ž G 2 ., using multiplicati¨ e notation in H 2 Ž k, 2 .. Žiii. e⬘Ž G . s e⬘Ž GrZ ., where Z is central in G. Živ. e⬘Ž G mk E . s ResŽ e⬘Ž G .., where E is a field extension of k, and Res: H 2 Ž k, 2 . ª H 2 Ž E, 2 . is the usual restriction map. Žv. e⬘Ž ResEr k GE . s Cor Ž e⬘Ž GE .., where E is a finite extension of k and Cor: H 2 Ž E, 2 . ª H 2 Ž k, 2 . is the usual corestriction map. We want to show that eŽ G . s e⬘Ž G .. It suffices to show that s ⬘. By Proposition 1, we only have to do this for G absolutely quasi-simple. PROPOSITION 2.
s ⬘.
Proof. This is just a question about coweight and coroot lattices. Indeed, the adjoint representation ad: ᒄ⬘ ª ᒐ ᒌ induces a map from the coroot lattice nr of ᒄ⬘ to the coroot lattice ⌳ r of ᒐ ᒌ. It also maps the
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WEE TECK GAN
coweight lattice nw of ᒄ⬘ into a lattice ⌳ that contains ⌳ r as a sublattice of index 2. ⌳ is simply the lattice corresponding to the group SO. Note that if dimŽ ᒄ⬘. is odd, then ⌳ is simply the coweight lattice. Otherwise, it is a lattice of index 2 in the coweight lattice. Hence, we get an induced map: nw rnr ª ⌳r⌳ r . This map of lattices induces the map ⬘ of finite group schemes, and so we denote it by ⬘ as well. Indeed, if nUw Žrespectively nUr . denotes the root Žrespectively weight. lattice, then the finite group scheme Z is the Cartier dual of the constant group scheme nUr rnUw . To compute this map of lattices, it suffices to assume that ᒄ⬘ is defined over ⺓, so that we can regard the lattices above as sitting in ᒄ⬘ and ᒐ ᒌ. So let H␣ , X␣ , Y␣ 4 be a Chevalley basis of ᒄ⬘. Using the isomorphism ᒐ ᒌ ( n2 ᒄ⬘ described in wF-H, p. 303x, we see that for any t g ᒑ, the Lie algebra of the maximal torus of G⬘, we have ad: t ¬
Ý
␣g⌽ q
␣ Ž t.
1 2
X␣ n Y␣ g ᒑ Ž ᒐ ᒌ . .
Let t␣ s 12 X␣ n Y␣ . Then the t␣ ’s lie in ⌳ and can be completed to a basis of ⌳. Furthermore, the intersection of the ⺪-span of the t␣ ’s with ⌳ r is generated by t␣ " t : ␣ ,  g ⌽q4 . Notice that if t g nw , then ␣ Ž t . g ⺪ for all ␣ , and so adŽ t . g ⌳. We can check if an element of nw lands in ⌳ r as follows. There is an augmentation map:
: ⺪ w t␣ x ª 12 ⺪ cs Ý c␣ t␣ ¬
1 2
Ý c␣ .
Then c lies in ⌳ r if and only if Ž c . g ⺪, i.e.,
: ⺪ w t␣ x r⌳ r l ⺪ w t␣ x ( 12 ⺪r⺪. Hence adŽ t . lies in ⌳ r if and only if 12 Ý ␣ g ⌽ q ␣ Ž t . g ⺪, i.e., if and only if Ž t . g ⺪. Hence ( ⬘ s < Z , as required. COROLLARY 3.
eŽ G . s e⬘Ž G ..
EXAMPLE. Suppose k s ⺢. Then we have the Cartan decomposition of ᒄ and ᒄ⬘: ᒄsᒈ[ᒍ ᒄ⬘ s ᒈ⬘ [ ᒍ⬘.
A NOTE ON KOTTWITZ’S INVARIANT
eŽ G .
377
The Killing form is positive definite Žrespectively, negative definite. on ᒍ and ᒍ⬘ Žrespectively, ᒈ and ᒈ⬘.. Let n s dimŽ ᒈ . and m s dimŽ ᒈ⬘.. Then we have w Ž BG . s Ž y1 .
1 2
n Ž ny1 .
w Ž BG⬘ . s Ž y1 .
1 2
m Ž my1 .
.
Since G and G⬘ are inner twists of each other, n and m must have the same parity. Hence, w Ž BG . rw Ž BG⬘ . s Ž y1 .
1 2
Ž nym .
s Ž y1 .
1 2
Ž di m GrKydi m G⬘rK ⬘ .
,
where K Žrespectively, K ⬘. is the maximal compact subgroup of G Žrespectively, G⬘.. Remarks. Ž1. This computation of eŽ G . for k s ⺢ seems to be simpler than that given in wKx. Ž2. It would be nice to show, using this definition, that if k is p-adic, then e⬘Ž G . is as computed in wKx. However, I am unable to prove this.
REFERENCES wF-Hx W. Fulton and J. Harris, Representation theory, Grad. Texts in Math. 129. wKx R. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 Ž1983., 289᎐297. wSx W. Scharlau, Quadratic and Hermitian forms, Grundlehren Math. Wiss. 270. wSex J. P. Serre, Cohomologie Galoisienne, Lecture Notes in Math. 5.