Hecke algebras and affine flag varieties in characteristic p

0.24 MB·English

by  Tobias Schmidt

HECKE ALGEBRAS AND AFFINE FLAG VARIETIES IN
CHARACTERISTIC p
5
1 TOBIAS SCHMIDT
0
2
Abstract. Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke
ct algebrawithcoefficientsinthealgebraicclosureFp ofFp. LetF betheaffineflagvariety
O over F associatedwith G. We show, in the simply connected simple case, that a torus-
p
equivariantK1-theoryofF (withcoefficientsinF )admitsanactionofH byDemazure
2 p
operators and that this provides a model for the regular representation of H.

]
T
R
.
h Contents
t
a
m 1. Introduction 1
2. Algebraic loop groups and their affine flag varieties 3
[

3. Equivariant K-theory 7
3
v 3.1. Schubert classes 7
5 3.2. Demazure operators 10
5
3.3. Iwahori-Hecke algebras 14
0
6 References 15
0
.
1
0
5 1. Introduction
1
: Let F be a p-adic local field with ring of integers o and let F be the algebraic closure
v F p
i of its residue field. Let G be a connected reductive linear algebraic group over F and
X
denote by GpFq the group of its F-rational points. For a choice of Iwahori subgroup
r
a I Ă GpFq, the corresponding Iwahori-Hecke algebra H over F equals the convolution
Fp p
algebra F rIzGpFq{Is of F -valued functions with finite support on the double cosets of
p p
GpFq mod I. The algebra H and a certain extension H p1q of it associated to the
Fp Fp
maximal pro-p subgroup of I - the pro-p Iwahori-Hecke algebra - were systematically
studied by Vign´eras [23], [24], [25]. Among their first applications is the mod p local
Langlands program for the group GpFq [6].
In this note we only deal with H and realize it as an equivariant algebraic K-theory of
Fp
its affine flag variety F over F . We view this result as a first step to analyze the mod
p
The author wouldlike to acknowledgesupport of the Heisenberg programmeof Deutsche Forschungs-
gemeinschaft (SCHM 3062/1-1).
1
2 TOBIAS SCHMIDT
p geometry of the algebras H and H p1q. In particular, we hope that the geometry of
Fp Fp
affine Springer fibres in F [5] will eventually shed a light on the representation theory of
these algebras, in analogy to the classical case of the finite complex flag variety [11]. We
also note that the occurence of affine geometry in the mod p setting can be viewed as a
consequence of the failure of the Bernstein presentation for the algebras H and H p1q
Fp Fp
[24].
To give more details about the present article, let us assume that G extends to a split
simply connected semisimple and simple group scheme over the integers. We denote its
base change to F by the same letter. Let T Ă G be a maximal torus. We choose an
p
Iwahori subgroup B Ă GpF rrtssq whose projection to GpF q contains T and let
p p
G “ GpF pptqqq{GpF rrtssq resp. F “ GpF pptqqq{B
p p p
betheaffineGrassmannianresp. theaffineflagvarietyofGoverF . Theyareind-schemes
p
whose structure can be defined through the increasing union of Schubert varieties S
w
indexed by elements w in the algebraic cocharacters Λ of T resp. in the affine Weyl group
W of G. By definition, S equals the closure of the B-orbit through w˚ and has a natural
w
structure as finite dimensional projective variety over F . Let T˜ be the extension of T by
p
the ’turning torus’ G (which corresponds to the derivation element d in the Kac-Moody
m
setting). Then T˜ acts on G and F and preserves the stratification into Schubert varieties.
Denote by K1 pS q the Grothendieck group of T˜-equivariant coherent module sheaves on
T˜ w
S and put K1 p¨q :“ K1 p¨qbF . We define
w T˜ T˜ p
K1 pFq :“ limK1 pS q
T˜ ÝÑ T˜ w
w
and show that the Schubert classes rO s provide an F rT˜s-basis for K1 pFq. Here, F rT˜s
Sw p T˜ p
denotes the ring of algebraic functions on T˜ over F . The analogous result holds for G.
p
Letting π : F Ñ G denote the projection, the pull-back
π˚ : K1 pGq ãÑ K1 pFq
T˜ T˜
is a linear injection whose image can be described explicitly. To each simple affine re-
flection s corresponds a smooth Bruhat-Tits group scheme P over F rrtss with connected
s p
fibres and with corresponding partial affine flag variety F . The projection π : F Ñ F
s s s
is a P1-bundle. One obtains a Demazure operator D :“ π˚π on K1 pFq in analogy to
s s s˚ T˜
the classical case of the finite flag variety treated in [3]. The operators D satisfy braid
s
relations and quadratic relations which are exactly the ones appearing in the Iwahori-
Matsumoto presentation of H . The latter is described in terms of the F -basis given by
Fp p
the characteristic functions τ of the double cosets IwI in the classical way [9], however in
w
our situation, the parameters satisfy q “ 0 for all s, since F has characteristic p. Letting
s p
τ act through D therefore turns K1 pFq into a right module over H “ H bF rT˜s.
s s T˜ FprT˜s Fp p
HECKE ALGEBRAS AND AFFINE FLAG VARIETIES IN CHARACTERISTIC p 3
In this situation, our main result says that mapping the function p´1qℓpwqτ to the Schu-
w
bert class rO s for each w induces an isomorphism
Sw
H Ý»Ñ K1 pFq
FprT˜s T˜
as right H -modules. Here, ℓ denotes the length function on W. Finally, specialising
FprT˜s
the isomorphism at the unit element T˜ “ 1 yields an isomorphism between H and the
Fp
absolute K1-theory K1pFq (with coefficients in F ) as right H -modules. In particular,
p Fp
any irreducible right H -module appears as a quotient of K1pFq. As a corollary, we
Fp
show that the restriction of π˚ to the module spanned by Schubert classes supported on
anti-dominant cocharacters Λ realizes the regular representation for the spherical Hecke
´
algebra F rGpo qzGpFq{Gpo qs viewed inside H via the mod p Satake isomorphism [7].
p F F Fp
We remark that all our results hold in fact with F replaced by any subfield k Ď F .
p p
We expect all results to extend to arbitrary connected and split reductive groups. More-
over, replacingtheIwahori-subgroupB Ă LGbytheinverse imageoftheunipotentradical
of the Borel subgroup of G in question should extend all results from H to H p1q. We
Fp Fp
leave these issues for future work.
To make a brief comparison with the complex case, we recall that over the complex
numbers the link between representations of the Iwahori-Hecke algebra and equivariant
K-theory was discovered by Lusztig [14] and occupies a central position in applications
to the local Langlands program. We only mention the proof of the Deligne-Langlands
conjecture by Kazhdan-Lusztig [11] where it is shown that the equivariant K-homology of
thecomplexSteinbergvarietyoftriplesaffordsanaturalHeckeactionthroughq-analogues
of Demazure operatorsandthat thisprovides a model for the regular representation of the
Hecke algebra. We also point out that the theory of Demazure operators on equivariant
K-theory of affine flag manifolds is well-established in the complex Kac-Moody setting,
due to the work of Kostant-Kumar [12] and Kashiwara-Shimozono [10].
Acknowledgements: TheauthorwouldliketothankElmarGroße-Klo¨nneandMarcLevine
for some interesting conversations related to this topic.
2. Algebraic loop groups and their affine flag varieties
We recall some definitions and concepts from [4], [17]. Let k be a field and let K “ kpptqq
be the field of formal Laurent series over k. Let G be a simply connected semi-simple and
simple group over k which is split over k. Let LG be its associated algebraic loop group,
i.e. the functor
R ÞÑ LGpRq “ GpRpptqqq
on the category of k-algebras. It is representable by an ind-scheme over k. Let L`G be
its associated positive loop group which is an affine group scheme over k representing the
4 TOBIAS SCHMIDT
functor
R ÞÑ L`GpRq “ GpRrrtssq.
Let B be the inverse image under the morphism pr : L`G Ñ G,t ÞÑ 0 of a Borel subgroup
in G. Thus, B equals a closed subscheme of an infinite dimensional affine k-space. The
affine Grassmannian G resp. the affine flag variety F of LG is the ind-scheme over k
G “ LG{L`G resp. F “ LG{B
representing the functor R ÞÑ LGpRq{L`GpRq resp. R ÞÑ LGpRq{BpRq. The natural
projection map π : F Ñ G is a proper and Zariski locally trivial fibration with fibers
L`G{B » G{prpBq (the split case in [17, 8.e.1]). The group LG acts on G and F by left
translations. Evaluating G on the structure map R ÞÑ Rpptqq gives an inclusion G Ă LG.
In this way, G and all its subgroups act on G and F.
For example, let G “ SL , the special linear group. Then LG equals the union of sub-
d
functors L G where L GpRq is the set of matrices M “ M ti of determinant 1 with
n n iě´n i
M a dˆd-matrix with entries in R. Each L G is representable as a closed subscheme of
i n
the affine scheme Ad2 and the inclusion L G Ă Lř G is a closed embedding. This
iě´n k n n`1
defines the ind-structure on LG. To make the ind-structure of G and F explicit, recall
that a lattice L ĂśRpptqqd is a Rrrtss-submodule of Rpptqqd such that there is n ě 0 with
tnRrrtssd Ď L Ď t´nRrrtssd
and such that the quotient t´nRrrtssd{L is a locally free R-module of finite rank. If
dL “ Rrrtss, the lattice is called special. We denote the set of special lattices over
R by LpRq and those for fixed n by L pRq. Then L is representable by a projective
n n
Ź
k-scheme. Indeed, the morphism
L pRq Ñ Grass pt´nkrrtssd{tnkrrtssdqpRq, L ÞÑ L{tnRrrtssd
n nd
definesaclosedembeddingintotheGrassmannianofnd-dimensionalsubspacesofthe2nd-
dimensional k-vector space t´nkrrtssd{tnkrrtssd. Note that multiplication by 1`t induces
a (unipotent) automorphism on the latter vector space which induces an automorphism
of the Grassmannian. The image of L equals its fixed point variety and hence, is indeed
n
closed. Each element of LGpRq gives rise to a special lattice in Rpptqqd by applying it to
the standard lattice Rrrtssd. Since GpRrrtssq equals the stabilizer of the standard lattice,
we obtain an isomorphism GpRq » LpRq. This defines the ind-structure of G in the case
of SL . Now a lattice chain in Rpptqqd is a chain
d
L Ą L Ą ¨¨¨ Ą L Ą tL
0 1 d´1 0
with lattices L in Rpptqqd such that each quotient L {L is a locally free R-module of
i i i`1
rank 1. Each element of LGpRq gives rise to a lattice chain with L special by applying
0
it to the standard lattice chain
L :“ Rrrtsse ‘ tRrrtsse
i j j
j“1,...,d´i j“d´i`1,...,d
à à
HECKE ALGEBRAS AND AFFINE FLAG VARIETIES IN CHARACTERISTIC p 5
where e ,...,e denotes the standard basis of Rpptqqd. Since BpRq equals the stabilizer
1 d
of the standard lattice chain, we obtain an isomorphism between FpRq and the space of
lattice chains L in Rpptqqd such that L is special. In this optic, the projection π is given
‚ 0
by sending a chain L to L and, hence, is a fiber bundle whose fibres are all isomorphic
‚ 0
to the usual flag variety G{prpBq of G. This defines the ind-structure of F in the case of
SL . For more details we refer to [5].
d
In the following, we are forced to work with a less explicit realization of the ind-structure
on F. To define it requires more notation. Our assumptions on G imply that Bpkq is an
Iwahori-subgroup of GpKq and hence, defines a fundamental chamber in the Bruhat-Tits
building of GpKq. We denote by B the associated smooth group scheme with connected
fibres over krrtss. Then B “ L`B in the notation of [17, 1.b]. Similarly, to any wall F
i
(i.e. facet of codimension 1) of this chamber, we let P be the smooth group scheme with
i
connected fibres over krrtsssuch thatP pkrrtssqisequal to theparahoricsubgroupof GpKq
i
associated to F . To P corresponds an infinite-dimensional affine group scheme L`P
i i i
over k with L`P pRq “ P pRrrtssq. The fpqc-quotient F :“ LG{L`P is representable
i i i i
by an ind-scheme over k, the partial affine flag variety associated to F . The projection
i
morphism
(2.1) π : F Ñ F
i i
is a principal L`P {L`B-bundle for the Zariski topology (the split case in [17, Thm.
i
1.4]). Moreover, L`P {L`B » P1, the projective line over k, according to [17, Prop. 8.7].
i k
We denote the reflection in the wall F by s .
i i
Let T bea maximal torusinG containedintheBorel subgroup prpBq andlet N pTqbeits
G
normalizer. Let W “ N pTq{T be the finite Weyl group of G with longest element w . It
0 G 0
may be viewed as the group generated by reflections in S :“ ts ,...,s u. Let Φ “ ΦpG,Tq
0 1 l
be the root system and let Φ` be the set of positive roots distinguished by prpBq. Let
Λ :“ X pTq denote the group of cocharacters of T. We view Λ as a subgroup of Tpkrrtssq
˚
via the mapping λ ÞÑ λptq. In particular, Λ acts on G and F.
Let W be the affine Weyl group of G. The group W is an affine Coxeter group with set
of generators S :“ S Y ts u. We denote by ď the Bruhat-Chevalley (partial) order on
0 0
W and by ℓp¨q the length function. Since G is simply connected, Λ coincides with the
coroot lattice whence W may be written as the semi-direct product W “ Λ¸W . When
0
considering Λ as a subgroup of W (whose group law is written multiplicatively), we will
use exponential notation.
Let P be either B or P or G and write for a moment F “ LG{L`P. Denote by
i P
˚ “ L`P P F the base point. Let WP be the set of representatives in W of minimal
P
length for the right cosets W{W where W Ă W equals the subgroup generated by 1 or
P P
s or S . The pB,Pq-Schubert cell C “ C pB,Pq is the reduced subscheme
i 0 w w
L`Bw˚ Ă F .
P
6 TOBIAS SCHMIDT
The pB,Pq-Schubert variety S “ S pB,Pq is the reduced subscheme with underlying
w w
set the Zariski closure of C in LG{L`P. It is a projective variety over k. Given u ď w
w
there is a closed immersion S ãÑ S and
u w
LG{L`P “ S
w
wPWP
ď
defines the ind-structure on F . If we let Fn be the (finite) union over the S with
P P w
ℓpwq ď n, then F “ Y Fn defines the ind-structure on F , too. In case of P “ P , we
P n P P i
shall write F “ Y Fn. In the following, we sometimes drop the base point ˚ from the
i n i
notation.
Let ∆ “ tα ,..,α u be the set of simple roots in Φ associated with S and let θ be the
1 l 0
highest root. Denote by Φaff the collection of affine roots pα,mq with α P Φ and m P Z.
Then α :“ p´θ,1q is the remaining affine simple root which induces the reflection s .
0 0
For each affine root pα,mq, there is an associated inclusion
φ : SL ãÑ LG
pα,mq 2
defined as follows [17, (9.8)]. The vector part α defines a k-morphism SL ãÑ G in the
2
usual way which induces by functoriality LpSL q ãÑ LG. The map φ equals the
2 pα,mq
composition with the morphism
a b a btm
SL ÝÑ LpSL q, ÞÑ .
2 2 c d ct´m d
ˆ ˙ ˆ ˙
The image of φ is contained in a Levi subgroup of the parabolic L`P Ă LG for i “
αi i
0,...,l. There is an irreducible k-linear representation V of the group scheme L`P whose
i i
pull-back via φ equals the unique irreducible representation of SL of dimension 2. Let
αi 2
LGˆL`Pi PpV q be the associated projective bundle over F . Let v be a highest weight
i i i
vector in V and for g P LG denote by rg,v s the class of pg,kv q P LGˆL`Pi PpV q. We
i i i i
then have a morphism of P1-bundles F Ñ LGˆL`Pi PpV q given by gL`B ÞÑ rg,v s.
i i
Lemma 2.2. The morphism F Ý»Ñ LGˆL`Pi PpV q is an isomorphism.
i
Proof. It suffices to check that the morphism is bijective on fibres [13, IV.4.2.E(3)]. By
LG-equivariance we may check this over the base point. However, the stabilizer of v in
i
L`P equals L`B and since V is irreducible, this implies L`P {L`B » PpV q. (cid:3)
i i i i
In [4, p. 54] Faltings constructs a central extension
1 ÝÑ G ÝÑ L¯G ÝψÑ LG ÝÑ 1
m
which acts on all line bundles on F (denoted L˜G in loc.cit.). The copy of G corresponds
m
to the central element c in the Kac-Moody setting. We consider T as a subgroup of
LG and consider its inverse image T¯ in L¯G, i.e. T¯ :“ ψ´1pTq. The central extension
has a unique splitting over L`B and we regard T as being contained in T¯. There is a
fundamental character ρ P X˚pT¯q whose restriction to T gives the character of V .
i i
HECKE ALGEBRAS AND AFFINE FLAG VARIETIES IN CHARACTERISTIC p 7
On the other hand, there is a natural identification
LGˆL`Pi PpV q » PpLGˆL`Pi V q
i i
and we have the family of line bundles Opnq for n P Z on this space. Each character
λ P X˚pT¯q yields an associated line bundle L “ LGˆL`B k on F [17, 10.a.]. Here, k
λ λ λ
denotes the one-dimensional T¯-representation induced by λ.
Lemma 2.3. Let 0 ď i ď l. The pull-back of Op´1q to F equals L . The L form a
ρi ρi
basis of the Picard group PicpFq of F.
Proof. Consider the two-dimensional L`P -representation V of weight ρ and highest
i i i
weight vector v P V as above. Consider the line bundle L “ G ˆL`B kv on F and
i i ρi i
fix a point gL`B P F. The fibre of L in gL`B equals rg,v s. Since the isomorphism
ρi i
F » PpG ˆL`Pi V q takes gL`B to rg,v s, we see that L equals the pull-back of the
i i ρi
tautological line bundle Op´1q. The second assertion follows from this together with [17,
(cid:3)
Prop. 10.1].
We need to introduce yet another torus. Let P be either B or P or G and abbreviate
i
F “ LG{L`P. For a given k-algebra R, its group of units a P Rˆ acts on the rings
P
Rrrtss and Rpptqq via tm ÞÑ amtm (’turning the loop’). This induces an action of G on
m
LG preserving the subgroup L`P. We obtain an action on F which commutes with the
P
action of T. We write
T˜ :“ G ˆT
m
for the extended torus. This copy of G corresponds to the derivation element d in the
m
Kac-Moody setting. The action of the extended torus preserves the Schubert cell C .
w
Indeed, we may write w “ w1eλ with w1 P W and λ P Λ. Given a P Rˆ one has
0
apw˚q “ apw1qapλptqq˚ “ w1λpatq˚ “ w1λpaqλptq˚ “ λpaqw˚ “ w˚
since λpaq P TpRq. Therefore T˜ indeed stabilizes C . It induces an action of T˜ on the
w
Schubert variety S . Even more, w˚ is the only fixpoint in C for the action of T and T˜.
w w
Namely, the cell C is isomorphic to a finite direct sum over copies of ’root spaces’ A1
w k
for T˜ indexed by affine roots pα,mq and with ’root vectors’ tmX . The isomorphism is
α
T˜-equivariant and maps w˚ to the origin in ‘A1. Here, X denotes a root vector for the
k α
root group U Ă G and the torus T˜ acts on tmX through the character pa,sq ÞÑ amαpsq,
α α
e.g. [4, p. 46/53].
3. Equivariant K-theory
3.1. Schubert classes. LetP beeither B orP orGandletF “ LG{L`P. Supposefor
i P
a moment that S Ă F is a T˜-stable ind-subvariety which is, in fact, a finite-dimensional
P
algebraic k-variety. We then have the abelian category of T˜-equivariant coherent modules
on S. We denote the corresponding Grothendieck group by K1 pSq. Let f : Y Ñ S
T˜
8 TOBIAS SCHMIDT
be an equivariant morphism to S from another T˜-variety Y. If f is flat, then pull-
back of modules induces a homomorphism f˚ : K1 pSq Ñ K1 pYq. If f is proper, there
T˜ T˜
is a push-forward homomorphism f : K1 pYq Ñ K1 pSq induced by p´1qjRjf [19,
˚ T˜ T˜ j ˚
1.10/11]. Note that if f is actually a closed embedding, then of course Rjf “ 0 for j ą 0.
˚
ř
Applying this to the structure map of S, the tensor product endowes the group K1 pSq
T˜
with a module structure for the group ring
ZrT˜s :“ ZrX˚pT˜qs “ K1 pSpec kq.
T˜
Let H Ď T˜ be a closed subgroup. Restricting the T˜-action to H induces a linear homo-
morphism
(3.1) ZrHsb K1 pFq ÝÑ K1 pFq.
ZrT˜s T˜ H
It is often bijective [15].
We define the Grothendieck group of F to be
P
K1 pF q :“ limK1 pFnq “ lim K1 pS q
T˜ P ÝÑ T˜ P ÝÑ T˜ w
n wPWP
in analogy with the case of the complex Kashiwara affine flag manifold (k “ C) treated
in [22, 2.2.1]. The transition maps here are induced by the push-forward along the T˜-
equivariant closed embeddings Fn ãÑ Fn`1 resp. S ãÑ S for u ď w.
P P u w
Let Un :“ FnzFn´1 with U0 “ Spec k. We let j : Un ãÑ Fn denote the corresponding
P P P P n P P
open embedding.
Lemma 3.2. There is a linear isomorphism K1 pUnq » ‘ ZrT˜s.
T˜ P ℓpwq“n
Proof. We have Un “ C with a linear T˜-action on C » Aℓpwq, cf. discussion
P ℓpwq“n w w k
at the end of previous section. So the claim follows from homotopy invariance of K1 [20,
š T˜
(cid:3)
4.2].
We have the element rO s in K1 pF q for w P WP. In the complex Kac-Moody setting
Sw T˜ P
these Schubert classes give rise to ’Z-bases’ in the corresponding K-theories of the affine
flagmanifold[10],[12]. Inthefollowing,weareratherinterestedincertainfieldsofpositive
characteristic p ą 0 where we are not aware of corresponding facts in the literature. The
following results have straightforward proofs and are sufficient for our purposes. We let
K1 p¨q :“ K1 p¨qb k and kr¨s :“ Zr¨sb k
H H Z Z
for any closed subgroup H Ď T˜. If H “ 1, we write K1 instead of K1 . We denote by F
t1u p
the algebraic closure of the finite field F with p elements.
p
Proposition 3.3. Let k Ď F . The krT˜s-module K1 pFnq is free on the basis given by
p T˜ P
the Schubert classes rO s with ℓpwq “ n. If H Ď T˜ is closed subgroup, then (3.1) is an
Sw
isomorphism after tensoring with k.
HECKE ALGEBRAS AND AFFINE FLAG VARIETIES IN CHARACTERISTIC p 9
Proof. We have ZrHsb K1‚ “ K1‚ for those H-varieties which have trivial H-action and
Z H
where K1‚ denotes ordinary K1-theory. Applying this to ‚ “ 1 and using
K1pSpec kq “ kˆ b k “ 0,
Z
the first equivariant K1-group of Un “ C thus vanishes upon tensoring with k.
P ℓpwq“n w
The localization sequence, tensored with k,
š
0 ÝÑ K1 pFn´1q ÝÑ K1 pFnq ÝjÑn˚ K1 pUnq ÝÑ 0
H P H P H P
is therefore exact [20, 2.7]. Since a preimage under j˚ of the unit vector at the w-th
n
position in ‘ ZrHs is given by rO s, the first assertion follows now by induction on
ℓpwq“n Sw
n with H “ T˜. The bijectivity of (3.1) follows now easily from this. (cid:3)
Corollary 3.4. Let k Ď F . The krT˜s-module K1 pF q is free on the basis given by the
p T˜ P
rO s with w P WP.
Sw
Remark: The ind-structure on F may also be defined via the pP,Pq-Schubert varieties
P
equal to the closure of the L`P-orbits in F [17]. Since an L`P-orbit is not a topological
P
cell anymore, we chose pB,Pq-Schubert varieties to calculate the equivariant K1-theory
of F .
P
For each n the set of fixpoints pFnqT˜ is a non-empty closed subvariety of Fn. We let
P P
ι : pFnqT˜ ãÑ F be the corresponding closed embedding. We have the following simple
n P n
version of the Thomason concentration theorem for fixed points varieties [21, Thm. 2.1].
Proposition 3.5. Let k Ď F . The variety pFnqT˜ equals the union of the finitely many
p P
points w˚ P F for ℓpwq ď n and w P WP. One has a linear isomorphism
P
ι : K1 ppFnqT˜q Ý»Ñ K1 pFnq
n˚ T˜ P T˜ P
which maps the class of the point w˚ to rO s.
Sw
Proof. One has
pFnqT˜ “ pC qT˜ “ tw˚u
P w
ℓpwqďn ℓpwqďn
ž ž
by the discussion at the end of the previous section. For the second assertion we consider
the cartesian diagramm
pUnqT˜ jn // pFnqT˜
P P
˜ιn ιn
(cid:15)(cid:15) (cid:15)(cid:15)
Un jn // Fn.
P P
Flat equivariant base change for equivariant K1-theory yields j˚ι “ ˜ι j˚ [19, 1.11]. We
n n˚ n˚ n
obtain a commutative diagram
10 TOBIAS SCHMIDT
0 // K1 ppFn´1qT˜q // K1 ppFnqT˜q // K1 ppUnqT˜q // 0
T˜ P T˜ P T˜ P
ιn´1˚ ιn˚ » ˜ιn˚
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // K1 pFn´1q // K1 pFnq // K1 pUnq // 0.
T˜ P T˜ P T˜ P
where the left-hand square commutes by functoriality for proper maps [19, 1.11] and the
right-hand isomorphism comes from homotopy invariance. The lower horizontal sequence
is exact by the proof of the preceding proposition and this implies the exactness of the
upper horizontal sequence. The result follows now by induction on n. (cid:3)
3.2. Demazure operators. We fix a number n and consider the preimage of Fn under
i
the projection π : F Ñ F from 2.1. Since π is flat and proper we have the homomor-
i i i
phisms
πi˚//
KT1˜pπi´1Finq oo KT1˜pFinq .
π˚
i
In analogy to the case of the finite flag variety [3] and the complex Kashiwara affine flag
manifold [10], we define a linear operator on K1 pπ´1Fnq by
T˜ i i
D :“ π˚ ˝π .
i i i˚
Lemma 3.6. The operator D extends to an operator on K1 pFq.
i T˜
Proof. Since π is a principal P1-bundle, each π´1Fn is a closed algebraic subvariety in
i i i
F and these varieties define, for varying n, the ind-structure on F. It suffices therefore
to show that D is compatible with the transition map K1 pπ´1Fn´1q Ñ K1 pπ´1Fnq. We
i T˜ i i T˜ i i
consider the cartesian diagram
π´1Fn´1 πi // Fn´1
i i i
˜ι ι
(cid:15)(cid:15) (cid:15)(cid:15)
π´1Fn πi // Fn
i i i
where the map ι is the natural inclusion and the map ˜ι is induced by it. Flat equivariant
base change for equivariant K1-theory yields π˚ι “ ˜ι π˚ and functoriality for proper
i ˚ ˚ i
maps yields ι π “ π ˜ι . Hence
˚ i˚ i˚ ˚
˜ι π˚π “ π˚ι π˜ “ π˚π ˜ι .
˚ i i˚ i ˚ i˚ i i˚ ˚
(cid:3)
We denote the resulting operator on K1 pFq by the same symbol D . As in [3, Prop. 2.6],
T˜ i
the projective bundle theorem in K-theory identifies the operator D as an idempotent
i
projector as follows. We note that under the isomorphism F » PpLG ˆL`Pi V q of 2.2
i