In 1968 Milnor [M] pointed out the remarkable similarity between the algebraic formalism of R-torsion in topology and zeta functions fi la Well in dynamical.
Inv. Math. N105, 185-216 (1991)
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mathematicae
9 Springer-Verlag1991
R-torsion and zeta functions for locally symmetric manifolds Henri Moscovici* and Robert J. Stanton*,** Department of Mathematics,Ohio State University,Columbus,Ohio 43210 1174,USA Oblatum 2 7 - X I I - 1 9 8 9 & 1 8 - X I I - 1 9 9 0
Introduction In 1968 Milnor [M] pointed out the remarkable similarity between the algebraic formalism of R-torsion in topology and zeta functions fi la Well in dynamical systems. From a different and purely analytical perspective, Ray and Singer [R-S2] exhibited an identity expressing the holomorphic analogue of R-torsion for surfaces (of genus > 1) in terms of classical Selberg zeta functions. They also conjectured the equality of R-torsion and analytic torsion, subsequently proved independently by Cheeger [C] and Miiller [Mii]. These developments pointed towards an intimate relationship between torsion invariants and dynamical zeta functions. This theme has been thoroughly investigated by Fried who devised for any smooth flow and any fiat bundle over the underlying manifold a certain formal zeta function counting the periodic orbits of a flow with appropriate multiplicities. He was able to show for a variety of flows ([FI, F2]) that the zeta function associated to any acyclic flat bundle is actually meromorphic on a neighborhood of [0,oe), regular at 0, and that its value at 0 coincides with the R-torsion with coefficients in the given flat bundle and thus is a topological invariant. Because of the analogy with the Lefschetz fixed point formula, Fried coined the term "flow with the Lefschetz property" in reference to such a flow. In particular, he proved that the geodesic flow of a closed manifold of constant negative curvature has the Lefschetz property IF3] and conjectured that this remains true for any closed locally homogeneous Riemannian manifold. In this paper we shall prove Fried's conjecture in what is arguably the most interesting case, namely for closed locally symmetric manifolds of nonpositive sectional curvature. The main novelty of our approach consists in the introduction of certain "supersymmetrized" zeta functions of Selberg type for higher rank locally symmetric spaces of non-compact type, which admit meromorphic 9 Supported in part by NSF Grant DMS-8803072 "" Part of this work was done while visiting the University of Chicago and Universit+ Louis Pasteur
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continuation and satisfy a functional equation. They play the same key role in the proof as the usual Selberg zeta functions do in the rank 1 case. We now proceed to state our result and outline the main steps of the proof. First, let us recall Fried's general definition of the formal zeta function associated to a flow [F2]. Let M be a compact, connected, smooth manifold, equipped with a smooth flow ~b which admits an invariant complementary distribution. The set of periods of closed orbits of q~ is discrete and the periodic set breaks up into connected components. Each such component C is compact, hence has a Fuller index, indv(C ) E Q. All orbits in C have the same period, dc, and are freely homotopic, thus determining a conjugacy class [Yc] in nl(M). Given a finite-dimensional representation r of nl (M), define a formal Dirichlet-type function by the formula
Z~(s) = exp - E Tr q)(~c)ind F(C)e -s~ . C
If M is the unit tangent bundle S X to a compact locally symmetric space X and 9 is the geodesic flow, then the connected components of the periodic set are parametrized by the non-trivial conjugacy classes [~] in F = nl(X ). Each connected component X~ is itself a compact locally symmetric manifold of nonpositive sectional curvature. ~ restricts to a periodic flow on Xr and the quotient
Xr = X T / ~ is a smooth orbifold whose Fuller index equals Z(2~7)/p~, where Z(.~) E Q is the orbifold Euler characteristic of A'7 and #~ is the multiplicity of a generic orbit of ~]X~. Given an acyclic unitary representation q~ of F, the above formal zeta function takes the form Z~(s) = exp - ~
Trr
~
.
I71#1
Our main result can be stated as follows: Theorem. Z,(s) is analytic for Re s >> 0 and can be meromorphically continued to
the entire complex plane with s = 0 a regular point. Moreover, Z,(O) = %(SX), where z~(SX) = Z~o(X)2 is the R-wrsion of the acycIic flat bundle defined by the unitary representation q~ o f n 1(X) -~ n 1(SX). A brief synopsis of the proof is now in order. First, one uses the CheegerMiiller theorem to replace the R-torsion by analytic torsion. Next, one passes from zeta-function determinants to heat kernels, i.e., in the formalism we expound in Sect. 1, to theta-function determinants. The trace of the heat flow is then decomposed, in Sec. 2, according to free homotopy classes, by means of the welt-known recipe of Selberg [Se]. A certain amount of work is required at this point to recognize the coefficients of this expansion as Euler characteristics of orbifolds (cf. Sec. 3). This is one of the new features distinguishing the higher rank case from the rank one and the fiat case, in which the periodic geodesics are either all isolated or only the isolated ones contribute to the expansion. The main difficulty however, comes from the lack of an adequate theory of Selberg-
R-torsion and zeta functions for locally symmetric manifolds
187
type zeta functions for locally symmetric spaces of higher rank. We finesse this impediment by constructing, in Sec. 4, certain "super" Selberg zeta functions, Z~(s), 0 _< # < 2m < dimX, as alternating products of formal Selberg-like functions, which reduce to genuine Selberg zeta functions only in the rank 1 case. Each function Z~(s) is related to a theta-regularized determinant hence is meromorphic on 113 and moreover satisfies a functional equation. Finally, Zr itself can be expressed as an alternating product of these, 2m
z~(s) = 1-I z~(s - m + OI - I / . ~=0
Not surprisingly, the functional equations play a crucial role in identifying the special value Zr with the R-torsion.
1 Operator determinants In this preliminary section we introduce a regularization procedure for determinants of Laplacians which is especially well suited for our purposes. For the sake of clarity, it will be convenient to axiomatize the spectral properties required by this procedure. So, throughout this section, A will denote a positive, possibly unbounded, self-adjoint operator on a separable Hilbert space, which is invertible with psummable inverse, i.e. (1.1)
TrA -p < oo for some p E [1,oo).
This implies that the associated theta-function
OA(t ) = Tre -tA,
t>0
is defined and moreover satisfies the estimate:
OA(t) = O(t -p) as t ~ 0 + . We impose the stronger requirement that it admits an asymptotic expansion of the form o~
(1.2)
OA(t) ~ ZC~kt~k
as
t--~ 0 +,
k=0
where ~k C ~ , - - p ~ ~Z0 ~ 51 ( " ' ' ( ~k ( ' ' ' and 0~k -'~ O0. On the other hand, one automatically has the "large time" estimate (1.3)
OA(t) = O(e -~1) as t - ~ oo.
where 21 is the first eigenvalue of A. From (1.2) and (1.3) it follows that 0A is a pseudofunetion [S, p. 38] and hence the finite part (in the sense of Hadamard) of its Laplace transform
(1 . 4 )
P f [ e-t~OA(t) dt 0
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is defined and meromorphic on ~i. Indeed, (1.4) can be written as a sum i
P f [ e-tXOA(t)dt + Tr [(A + 2)-% -(A+i)] 0
with the first term an entire function and the second meromorphic, with simple poles at 2 = --2~, k __> 1. Here 0 2m + ]A + [. If we expand e ta = ~, ~5-, we get rn
vam-2JP2j&Meta j=0
=
.
A m--j
~'~ ( t ' / 2 v ) e ( ~ - J ) p E j ~ G
M + o(tmv2r
j=0 with
2~ < m.
Substituting this into the integral and letting t ~ 0 +, one gets
kt(e) ~ t -C~
Am-JP2J~M{(X~+- ~(a_)' -AM,CXX(--I)/Atn}e --
--09
v2m_2Je_V2
dv . []
R-torsion and zeta functions for locally symmetric manifolds
201
Corollary 2.10. One has ct~
(9~kk~(e)= y'(--1) ~ / e-t(v2+(m-d)2)
(i)
-oo
(:c Moreover ~-,(_l)kk Tr e_tZx~o,k ~ t~/--c 7,
t ~ O+"
(ii)
Proof. It suffices to take (2.11) and use Lemma 2.5 to obtain (i). The assertion (ii) is immediate from Proposition 2.9 and standard estimates for the contributions from Lemma 2.7 from other F conjugacy classes to (2.1). [] We summarize the results of this section with the expression of ~'(-1)kkTr e-tZx~,k in group theoretical terms. As in [M-S] let o~1(F) denote the F-conjugacy classes of non-trivial elements in F contained in a fundamental, R-rank one Cartan subgroup of G.
Theorem 2.11. Let X be a compact locally symmetric manifold of odd dimension and F a flat vector bundle over X associated to a unitary representation (tp, F) of F = ul(X). Let A~, k be the Laplacian on k-forms with values in F. Then (2.12) O(t) = Z ( - - l ) k k Yr e-tZx~,k
=
~ [~]e~l(r)
[Gr : Go]-1 vol (GJFr)
Trip(y)~e(7-)~ [ - 1 - - ~
_ %1
~eP~
e-t~/4t Z ( _ l ) E + l e - t (m-d): (4rot)l/2 Z ai, r Z
e(w)ff)r(w(2i + OM))r
W~(M~) oo
+ dim Fvol X Z ( - 1)e / e-t (~2+(~-e)2) -oo
(,eg2~(M~)dim [We | (A~ p , ~ - AeVpm)|162
(Tt)
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H. Moscovici and R.J. Stanton
3 Euler characteristics
The goal of this section is to evaluate in group theoretical terms the Euler characteristics of the orbifolds arising from the periodic geodesics. This will enable us to express the Laplace transform of the torsion theta function O(t), discussed in the preceding section, in purely dynamical terms. We begin by reviewing briefly some basic facts about the geodesic flow of a locally symmetric manifold. A thorough discussion of this subject can be found in [D-K-V, w5]. Then, we outline some results from [M-S, w5] related to the disconnectedness and reductive nature of G~. These allow us to construct an algebraic descent to an equirank group that facilitates the evaluation of the Euler characteristics. The set of periodic orbits of the geodesic flow 9 on S X consists of a disjoint union of connected smooth submanifolds parametrized by conjugacy classes [7] 7~ [e] in F = hi(X). For each [7] the corresponding submanifold X 7 is diffeomorphic to F~ \ Gr/U~, U r a maximal compact subgroup of Gr. As F is co-compact each 7 is semisimple and so we may assume it has been put in standard position relative to the original choice of K. Then X~ ~ F~ \ G~/Kr and we write 7 = 717• with 7~ = exp Y~Y E exp p where u E p is a unit vector. The restriction of the geodesic flow to each X r can be described, in this picture, by q)s(FrgK~) = F r g e x p s Y K ~,
g E Gr.
The orbits of @lx~ are periodic geodesics in the free homotopy class corresponding to [7]- They all have the same length (7 and one has 7~ = exp {r Y" For most of the ensuing discussion we shall assume that [y] E gl(F) and that 7 = T t e x P [ r Y E AtAR = B. The group G~ is reductive, possibly disconnected. Both these features introduce technical complications. Let Go be the component of the identity of G~, and let C~ be the center of G~ Notice that since B is connected, B _c G0, in particular 7 E C~. Also it is easy to see that Cv is a normal subgroup of G~ and hence F~ (1 C r < F< Lemma 3.1. Let [7] ~ gl (F). Then, F~ A Cr is free abelian of rank I.
Proof This can be proved as in Lemma 5.3 (ii) of [M- S]. Lemma 3.2. Let [7] E d~ and we may take 71 in B.
[]
and let 71 be a generator o f F r n C < Then [71] ~ d~
Proof 71 ~ C~ = center of G~ which is contained in every Cartan subgroup of G~ It follows that h E B. As h is in F r ~ F and h is in B, one has [h] E gl(F). [] Definition 3.3. Let [ j E gl(F). One may pick a generator 7* of F~ N C~ such that
7 = (7") "~ with mr > 1. It follows from Lemma 3.2 that [r = r n / < .
R-torsion and zeta functions for locally symmetric manifolds
203
Returning to G~ we note that A~ is the split component of G~ Also Ct = C~?AR with C~ a compact abelian, possibly disconnected, group. Let G'~ denote the semisimple group (with finite center) such that Gt = G'~C~ Set r~ = G'~ n r~c~. Then r'~ is a discrete, co-compact subgroup of G'~. and, if necessary, we may take Or'~ ,ar~ and ~ torsion-free and of finite index in F'~. Set OFt = r~ A~ F;C~. Then OFt is normal in F~, torsion-free and of finite index in F t. Set ~ =o Ft\Gt/Kt and ~ =o F'~\C,',~/K'~,. Then S ~ acts freely on ~ ([M-S, Lemma 5.4]) with quotient space ~ Let )~(~ '~,)denote the Euler characteristic of ~ 't" Lemma 3.4. z(~ '~)r [r~ :0 r~l-t = c~-~lwr
I-I (e~, c~)[G~ : 6~ - ' vol (G~/F~). ~eP~
Proof We shall reduce the computation of z(~
to an equirank group situation and then use results from [H-P] and [Sc]. As usual MA~N is the cuspidal parabolic associated to B. Since 7 = 7i exp [~ Y E B, ~'t is an elliptic element in M ~ Notice that for such B we have M~r =- (M~t)~ = (G'~)~176 t and My, = G'tC~I. M (resp. KrIM 0 ) denote the maximal compact subgroup of Mr~ (resp. Let K7I MOt) relative to the restriction to M of the Cartan involution of G. Then
MtI/K~ ~-- --'i M~ "--~,/zM~and OF t' \ Mr, / K ~ =--~X; is a finite cover (of order [0Ft, :o F t, N G't~
of ~
N M~t~\--~tM ~ ,__rl/K M~ Although we have not produced
a discrete, co-compact subgroup of M ~ nevertheless the elliptic element 7/
and the discrete co compact group 0r; o Mo places 0r,t n Mo \
in
the local setting of [H-P]. So following their argument (p. 220-225) we take the orientable cover ~ --' X~ = oF~\G~/K~ o--, , , '~176 Thenz(~ X~)/[K t, : K ~,o] and z(~ can be obtained by orienting 0 ~,y then evaluating the Euler class on the fundamental cycle. t t Let Art be a maximal torus in K and set Y~t -_ 0 r t\G~/A~l. Then for a choice of orientation c(l-I~ce~ c0 is the Euler class of Y '~ and
Z(~
= IW(K~O/A~I)I"
Schmid ([Sc, p. 38]) evaluated c(rI~ep ~ e)[Y'~] obtaining (2n)-#P~(-1) #P~,n
v(0 F ~,' \ G t)t ]W~Z(G'~)I1-I,ep~(Q~,e). The choice of Haar measure in [Sc] is different from the one here with the relationship being v (~
\ G'~) = vol (~
\ G '~)2-~/2v(K '~)v ( A ~ ) - ' .
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H. Moscovici and R.J. Stanton
After these substitutions we obtain
z(~
t = [g tv :g~t O ]- 1 IW(Kv/Ar)I
--1
x (--1)#e~,"(2u)-#er2-~r/2v(K '~)v(Ar)-lvol (~ x
l \ Gv)
IW~:(G'e)IH (ov,~). 9EP7
A comparison with (2.9), together with the standard facts [K'~ : K'~ ~ = [G'~ : G'~~ = [ ~ 99ay], o IW(K ~~ = IW(a'~~ = IW ( a J h r ) l give z(0X,) = c~-ivol (0F7, \ a~)lWr ,
I-I (Ov, c~).
Now using the S ~ fibration 0X~ _, 0Xv,, and integration along the fiber (Lemma 5.8 in [M-S]) we get vol (~
\ G'~) = vol (~
\ Gv)/ - e e results, ec being a lower bound for the positive eigenvalues of the A~ 't. To obtain (4.4) we begin with (4.3) and make various identifications. From
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H. Moscovici and R.J. Stanton
Proposition (3.4)
z(2,)~, = [a, : ~]-'vol (r, \ 6,)c;-'lWe(6,)l l-I (e,, :). ct~P,,
Also P~ = P~v,I U Pc, ~q(7) = ~OM(',')~Or and ~er I l a c p ~ ; [ 1 - ~_~(~/)] = ~Qr Ad?ln). And as B is fundamental and split rank one, with 7 = 71exp fy Y, one has ~er = emt'r Then the result follows from the identity
e-Ze'~ j e-t~/4t 2z = e-Z2t(~i72 dt,
(4.6)
R e z q~ (--o0,0]
0
and an interchange of integrals. The justification for the interchange proceeds exactly as in the proof of Proposition 6.1 [M-S] and consequently we only sketch it. Fix T, 0 < T < o % a n d s e t T
l r = P f j 2(s + [m - El)Oe(t)e-t(s2+2"lm-tl) dt 0
and oo
loo = f 2(s + [ m - U[)Ol(t)e-~ (s2+2s]m--dl)dt. T
Consider first I~ = Tr,(2(s + Im - EJ) (A~ + s 2 -}- 2slm - dl) -1)
e -T(/x~+s2+2slm-dl).
Since Re (s2 + 2sire- dl) > 0 the kernel of this operator is in c6 I(G) and the trace can be computed by orbital integrals. From Kuga's formula and the argument in [M-S, p. 660], one sees that this kernel is also a pseudocusp form as in Lemma 4.2. Thus, we get the absolutely convergent series e m f~,
Z~
Trq~y))~t~7)2~ ~ - ~ det (I - AdT[.) D~(~)
[?]C ~1 (F)
(4.7) j
2(s + Im -- fl) eiVt~,e-T(v2+(s+lm-fl)2) dr. v a + (s + I m -- El) 2
-oo
For IT, if we ignore the contribution from the identity element, as we may, we have a convergent integral. We can then proceed as on p. 661 getting the absolutely convergent
(4.8)
Z ['/]ESt (F)
)~Tem&D r
Tr~p(~)Z(27)de t ( I - Ad?[~)
T
e_~/4 t
f 2(s + [m--[[)e -t(s+lm-r (4nt)l/~ dt. 0
R-torsion and zeta functions for locally symmetric manifolds
211
Adding (4.7) and (4.8) and undoing the Fourier transform in (4.7) yields the expression
Z
e_(2/4 t
2(s +Im + dl)e -t(s+lm-el)2 (4~t)1/2 dt
27em& Dl(7)
Tr ~~
det (I -- Ad 71,)
[?]Cgl(F)
0 ~3
+ P f 2(s + ]m -- d[) { e-t(sa+Zslm-tDf[(e) dt. 0
The integral is evaluated using the identity (4.6), completing the proof.
[]
We are now in a position to introduce the super Selberg zeta functions. Definition. For z E 9 satisfying Re z 2 > (m - d) 2 -- ee and Re z q~ ( - ~ , 0] define logZ~(z) = -
De (7) e-(z-m)& ~ Tr,.p(7)),(2.j det ( i - AdTI.) #~ ; [7]#1
recall that/2 7 is the generic multiplicity. Note that by Remark 3.7 only classes [7] c d~I(F) contribute to the sum. Moreover, this series converges absolutely and uniformly on compact subsets, as follows from the previous proposition and the fact that the length spectrum is bounded away from zero. For the same reason, one can see that limz_.+~ log Z~(z) = O.
Remark. The function Z~(s) reduces to an ordinary Selberg zeta function only when rank G/K = 1. For higher rank spaces, Z~(s) can be viewed as an alternating product 2k
Z~(s) = l-I z~d(s) (-l~j,
2k = dimp,,
j=0 where logZ~d(s) can be defined, at least at a formal level, by a sum reminiscent of the expression for a rank 1 Selberg zeta function, but involving all conjugacy classes [7] in F, [7] # {1}. Thus, the formula for Z~(s) incorporates hidden cancellations which remove the contribution of the conjugacy classes outside gl(r). Theorem
4.8.
Z~(z) = exp - ~
[jqq
De(y ) e-(z-m)e~ Trrp(jZ(2Jdet(iCXdTi, ) U~
is holomorphic if Re z 2 > (m - d)2 ~e and Rez ~ (--o%0]. Moreover Z~ has a meromorphic continuation to C given by m k ~ z2k+l --2nc ~ . ( - - 1 ) a k
r - det ~(A~)e e Z~(z) -(c = dim Fdim X) ; (i)
k=0
det 0(I + (z 2 -- (m _ y)2)A~e-!),
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H. Moscovici and R.J. Stanton
satisfies a functional equation m k E z2k+l 4nC ~ ( - - 1 ) a~ 2k+l
Z~(--z) = e k=O Z~(z); has the product formula (iii) Z~(z)Z~(-z)t r = [det~(/xC~)deto(I + and satisfies the duality relation (iv) Z~(z) r = Z~2m-g(z). Proof From (4.4) we have (ii)
(z 2 -- (m - f)2)/x~-l)]2;
~3
a
log Z~o(z r ) = 2zTr ~
(m -
+ z2-
y)2)-1 _
pf 2z / e-tt~Z-(m-:?)ff(e) dt ,I 0
while (1.5) gives d
logdet0(I+(z 2-(m-f))A 2
2zTr 0 ( A ~ +
t~-I ) =
Z2
-(m-#)2)-1.
Thus we get
logZ~(z)=c+logdeto(i+(z2_(m -
f) 2)/~ f-l))
oo
+ pf f e_t(z2_(m_t)2)ff(e) dt t 0
To prove (i) we evaluate the Pf integral and determine c using limz~+oo. Using (iii) of L e m m a 4.5 we compute
dt Pf /et(~2 - - (m- 02)ff(e)dt t = Pf f e-tZ2c~ dkF ( 1k)+ ~ tk+3/2 0
0
k=0
=e~[dkF(k+~)F(--k--~)z k=0 m
~2k+l
= - 2 . c Z (-llka
,
k=0
and we note the small time asymptotics of
Oe(t)
m
Oe(t) "~ ce-t('-r
1
x-~ e F ( k + 7) _
2-, ak t ~ 5
Zt~.c~,
k=0
where a n = n + ~, n __> --m - 1, and
c. =
Z j-k=n+l
ca 'r k +
1\
9(m - -
{)2j
'
2k+l
R-torsion and zeta functions for locally symmetric manifolds
213
o f course j > 0 and m > k > 0. In the Stifling type estimate (1.12)' we set 2 = z 2 - (m - :)2 and e x p a n d [1 -- ( . ~ 2 ] ~ ,
obtaining
log det 0 (I + (z 2 - (m _ :) 2) A :-1) - ~-~(z 2 - (m - :)2)-~"F(%)c, - log det r
1 n=>-m- i
~ (m - : ) 2'
log det r(/',~).
r_>0
Substituting for c, and s u m m i n g over j, k we get log det0(I + (z 2 -- (m-- :)2 ) / ~ : - 1 ) -logdet r m
~e~2j+2r -- v !
k=O
--2j~2r F
(,+, "
j,r>O
= -- log det r (A~)
--~z+2k+'ca~r(k+ ~z-2U(m-:)2~r u-kk=O
u=0
(--W =
1 v! (u -
v)! "
But the sum over v is zero unless u = 0, hence l o g d e t 0(I + (z 2 -- ( m - :) 2 ) ~
:-1
)
~--logdetr k=0
As z ~ 4-oo, logZ~(z) --* 0 a n d hence c = l o g d e t ( A ~ and the p r o o f o f (i) is complete. There is a shorter p r o o f of (i) using the approximation (1 --x) -~" ~ 1 4- O(x), x -+ 0 + but the one given shows that logZ~ vanishes to oo-order at oo. T h e functional equation (ii) follows immediately from the form of the exponential factor and the z 2 in the 0-regularized determinant. Also (iii) is obvious from (ii). F o r (iv) we note that as representations of M, A % and A 2 " - e n are equivalent hence De(V) = Dzm_:(?). [] L e m m a 4.9. Let h = hI hR E B, h ~ e, Then
(4.9)
~h{~'aM Z(-a):z^:.}(h)
det(l-Adhl.) = lWC(6;h)I l-[ (0h.~)~,(h) l-I [1-C~(h)]" u,CPh
~t~-P~, I
Proof When h is regular, ~h = 1. T h e n (4.9) is the well-known (4.10)
det (I -- Adh[,) = ~ - ~ ( - 1 ) t T r (AeAd hi,).
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H. Moscovici and R.J. Stanton
If we multiply (4.10) by ~OM'Z~M, then (4.9) results essentially from L'HSpital's rule. Indeed since B is fundamental, split-rank one and h % e, we have A(gh) consists of imaginary roots. Then the only non-zero contribution to Nh{~0M'AMdet (I -- Ad (')l,}(h) is det (I - Ad hl,)~aM (h)~hgAM)(h). But ~h{ H
[1-~_:(-)]}(h)=Z
H
[1 -- ~_~(h)]
(~'a~) H
where a runs over the permutation group on Ph3" In [Sc, (5.14)], we find
a ctEPh,I
otEPh,I
Lemma 4.10. Let h = h t e x p r Y E B, h ~ e. Then ~[(--1)%re ~i ai't y" g(W)OOh(W(~i (4.11)
det (I - Adh],) =
~lWr
H
"~
QM))~w"~'(hl)
(Qh,~)~M(h) H [1-~_~(h)]
~t~-Ph,l
ctEP~,1
Proof Recall the notation from Lemma 2.8, ZAdn :
Z ai, {Z2` i
as representations of M +. As observed before Ad (exprY)l, = erId. Then (4.11) follows from (4.9) and the Weyl character formula. [] We are now able to prove the main result that relates R-torsion to a special value of a zeta function9 Theorem 4.11.
Z~(s) = e x p -
~
e-S gy
Trcp(~)z(X,) ~7
is an analytic function in the region Re s > m and Re (s - m) 2 ~ m2, Moreover 2m
Ze(s) = H Ze(s -- m
+ ~e) (-1)c"
"
~'=0
in particular Ze admits a meromorphic continuation to • with s = 0 a regular point. Also
z~(O) = ~(0) 2. Proof For Re s sufficiently large we have 2m
E(--1) t log Z g ( s
-- m +
E)
d=O
=--
~ Trq@)det(I--AdTlO [~legl(r)
/~
9
~e=o
(-1)%(2m-e)6Dc(~)
.
R-torsion and zeta functions for locally symmetric manifolds But as n o t e d
before
De(?) =
D2,,_i(7) , so the expression
215 in brackets
is
~ = o ( - 1)crierDe (7). However, f r o m (4.11) we see t h a t this is just det (I - A d ? I,), hence we o b t a i n the c o n v e r g e n c e for each in the stated d o m a i n . Z~ a n d the regularity Finally, using (iv) a n d
series for logZ~(s). A n e x a m i n a t i o n o f the d o m a i n o f s u p e r Selberg zeta f u n c t i o n shows t h a t each is c o n t a i n e d T h e m e r o m o r p h i c c o n t i n u a t i o n follows f r o m t h a t o f e a c h o f s = 0 f r o m the acyclicity a s s u m p t i o n . (iii) of T h e o r e m 4.8 we o b t a i n 2m
z~(o) =
1-I z ; r - m) e=o 2m 1-I Z~([ m - a l l ) C - l ) '
= fiZZ( -Irn-CI)(-l/ e=O
e=m+ ! m--I
= h z:(-Ime=0
II r
nl-I
= l - I [det ~(A~)]t-l)edet ~(AT) (-1)m e=o 2m = Y I [det ~(Ag)]( -1)e e=0 =~ff(0) 2. Here, the last equality follows f r o m L e m m a 4.3.
[]
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