数理解析研究所講究録
第 1826 巻 2013 年 168-179

168

Parabolic Reduction, Stability and the Mass
I. Special Linear Groups
Lin WENG

Abstract: Basic formulas exposing intrinsic relations on volumes of fundamental
domains, that of their stable portions and various zetas are obtained for special linear

groups, via theory of Eienstein series and parabolic reductions. Parallel theory forbundles over curves on finite fields is reviewed as well. Based on all this, conjectures

for general reductive groups are formulated.

1

Number Fields

1.1

Siegel’s Volume Formuladefined over , there are 3 naturally assoFor special linear group, its maximal compact
ciated groups, namely, the real Lie group. It is well known
subgroup
and the full modular groupmay be interthat the double quotient spacepreted as the space of isometric classes of rank lattices of volume one. Indeed, the metrics onin the Euclidean spaceare parametrized, and up to-equivalence, theby matrices
withmetric is uniquely determined by . As such, then the lattice structures
are finally determined modulo the automorphism groupof
Denote by
the moduli space of all full rank lattices inofvolume one. The above discussion exposes the following$SL_{n}$$\mathbb{Q}$$SL_{n}(\mathbb{R})$$SO_{n}(\mathbb{R})$$SL_{n}(\mathbb{Z})$$SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R})$$n$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$A\cdot A^{t}$$A\in GL_{n}(\mathbb{R})$$O_{n}(\mathbb{R})$$A$$SL_{n}(\mathbb{Z})$$\mathbb{Z}^{n}.$$\mathbb{R}^{n}$$\mathbb{M}_{\mathbb{Q},n}[1]$Fact 1. (Arithmetic versus Geometry) There is a natural one-toone correspondence$SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/SO_{n}(\mathbb{R})\simeq \mathbb{M}_{\mathbb{Q},n}[1].$, we may ask what is the
Associated the natural measure on
corresponding volume of the above space. Surprisingly, while the space, or the same,, is highly non-abelian,$SL_{n}(\mathbb{R})$$SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/SO_{n}(\mathbb{R})$$\mathbb{M}_{\mathbb{Q},n}[1]$169
or the same, non-commutative, according to Siegel, its volume can be
expressed in terms of the special values of Riemann zeta function, which
is abelian in nature.
Fact 2. (Siegel) (Volume of Fundamental Domain)$m_{\mathbb{Q},n}:=Vol(\mathbb{M}_{\mathbb{Q},n}[1])=\hat{\zeta}_{\mathbb{Q}}(1)\hat{\zeta}_{\mathbb{Q}}(2)\cdots\hat{\zeta}_{\mathbb{Q}}(n)$where$\hat{\zeta}_{\mathbb{Q}}(s)$denotes the complete Riemann zeta function and$\hat{\zeta}_{\mathbb{Q}}(1):={\rm Res}_{s=1}\hat{\zeta}_{\mathbb{Q}}(s)$1.2.

Stability

Among all lattices, motivated by Mumford’s fundamental work in algebraic geometry, we independently introduced the semi-stable lattices
in our studies of non-abelian zeta functions. By definition, a lattice iscalled semi-stable if for all sub-latticesof$\Lambda$$\Lambda$$\Lambda_{1}$$Vol(\Lambda_{1})^{rank\Lambda}\geq Vol(\Lambda)^{rank\Lambda_{1}}.$

Denote by
the moduli space of ranksemi-stable lattices of$vo$ lume 1. One checks thatis a closed compact subset ofWith the induced metric, define$n$$\mathbb{M}_{\mathbb{Q},n}^{ss}[1]$$\mathbb{M}_{\mathbb{Q},n}[1].$$\mathbb{M}_{\mathbb{Q},n}^{ss}[1]$$m_{\mathbb{Q},n}^{ss}:=Vol(\mathbb{M}_{\mathbb{Q},n}^{ss}[1])$A natural question is what is the volume

1.3.

$u\mathbb{Q},n.$High rank non-abelian zeta functions

Similarly, denote bylattices and bydecomposition

the moduli space of ranksemi-stable
its volume
part. Then we have a natural$n$$\mathbb{M}_{\mathbb{Q},n}^{ss}$$\mathbb{M}_{\mathbb{Q},n}^{ss}[T]$$T$$\mathbb{M}_{\mathbb{Q},n}^{ss}=\bigcup_{T\in \mathbb{R}>0}\mathbb{M}_{\mathbb{Q},n}^{ss}[T].$Easily one checks that there is a natural isomorphism$\mathbb{M}_{\mathbb{Q},n}^{ss}[T]\simeq \mathbb{M}_{\mathbb{Q},n}^{ss}[T’] \forall T, T’\in \mathbb{R}_{>0}.

(*)$

Using the above measure on
and the invariant Haar measure, we obtain a natural measureonon$\mathbb{M}_{\mathbb{Q},n}^{ss}[T]$$\frac{dT}{T}$$\mathbb{R}_{>0}$$d\mu$$\mathbb{M}_{\mathbb{Q},n}^{ss}.$170$i=0,1$ for
Moreover, there is a genuine cohomology theory
lattices over number fields $F$ for which the arithmetic analogue of the
duality, the Riemann-Roch theorem, the vanishing theorem holds. For
details, please refer to [W]. In the case of$h^{i}(F, \Lambda),$$\Lambda$$F=\mathbb{Q},$$h^{0}( \mathbb{Q}, \Lambda)=\log(\sum_{x\in\Lambda}e^{-\pi\Vert x\Vert^{2}})$

which was introduced earlier in [GS]. Denote by
the Arakelov degreeof , which overis simply $-$ log Vol . Following [W], define theassociated $mnkn$ non-abelian zeta functionby$d(\Lambda)$$\Lambda$$(\Lambda)$$\mathbb{Q}$$\hat{\zeta}_{\mathbb{Q},n}(s)$$\hat{\zeta}_{\mathbb{Q},n}(s);=\int_{\mathbb{M}_{\mathbb{Q},n}^{ss}}(e^{h^{0}(\mathbb{Q},\Lambda)}-1)\cdot(e^{-s})^{d(\Lambda)}d\mu,$${\rm Re}(s)>1.$

Then using the basic property of the above cohomology theory fornamely the duality, the RR and the vanishing theorem, tautologically,
we have the following$h^{i\prime}s,$Fact 3. (Weng) (0) (Relation with Abelian Zeta)$\hat{\zeta}_{\mathbb{Q},1}(s)=\hat{\zeta}_{\mathbb{Q}}(s)$;(i) (Meromorphic Extension)
is a well-defined holomorphic
function in ${\rm Re}(s)>1$ , and admits a unique meromorphic extension to
the whole -plane;(ii)
nctional Equation)$\hat{\zeta}_{\mathbb{Q},n}(s)$$s$$\infty$$\hat{\zeta}_{\mathbb{Q},n}(1-s)=\hat{\zeta}_{\mathbb{Q},n}(s)$(iii) (Singularities)
at $s=0,1$ . Moreover$\hat{\zeta}_{\mathbb{Q},n}(s)$;has only two singularities, all simple poles,${\rm Res}_{s=1}\hat{\zeta}_{\mathbb{Q},n}(s)=m_{\mathbb{Q},n}^{ss}:=Vol(\mathbb{M}_{\mathbb{Q},n}^{ss}[1])$In particular we see thatis naturally related to the special valueof the non-abelian zeta function.
$m_{\mathbb{Q},n}^{ss}$$\hat{\zeta}_{\mathbb{Q},n}(s)$1.4Parabolic Reduction: Analytic Theory

The high rank zeta functions are closely related with Eisenstein series.
In fact, we have

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Fact 4. (Weng) (i) (High Rank Zeta and Eisenstein Series)$\hat{\zeta}_{\mathbb{Q},n}(s)=\int_{\mathbb{M}_{\mathbb{Q},n}^{SS}[1]}\hat{E}(\Lambda, s)d\mu=\int_{(SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R}))^{ss}}\hat{E}^{SL_{n}/P_{n-1,1}}(1, g;s)$Heredenotes the complete Eisenstein series associated to thelatticedenote the relative (complete) Eisenstein seriesoninduced from thefunction 1 on the Levi factor of the
maximal parabolic subgroup and $P_{n-1,1}$ denotes the standard pambolic
subgroup of
corresponding to the partition $n=(n-1)+1$ , andthe part corresponding to the semi-stablelattices via Fact 1, which for our convenience will also be viewed as a;subset of(ii) (Analytic Truncation versus Arithmetic Truncation)$\hat{E}(\Lambda, s)$$\Lambda,\hat{E}^{SL_{n}/P}(1, g;*)$$con\mathcal{S}tant$$SL_{n}(\mathbb{R})$$P$$SL_{n}$$(SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R}))^{ss}$$SL_{n}(\mathbb{R})$$\Lambda^{0}1=\chi(SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R}))^{ss}$Namely, Arthur’s truncation of the constant function 1 ischaracteristic function of the subset
sisting of semi-stable points.

$simg_{s}^{\iota_{y}}$$(SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R}))$

the
con-

This is a number theoretic analogue of a result of Laffourge on the relation between analytic truncation and arithmetic truncation for function
fields.

Consequently,$\hat{\zeta}_{\mathbb{Q},n}(s)=\int_{SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R})}\Lambda^{0}\hat{E}^{SL_{n}/P_{n-1,1}}(1, g;s)$.

Heredenotes the Arthur’s truncation of the Eisenstein series $\hat{E}^{SL_{n}/P_{n-1,1}}(1, g;s)$ . On the other hand, by Langlands’ theory of Eisenstein series, we know that$\Lambda^{0}\hat{E}^{SL_{n}/P_{n-1,1}}(1, g;s)$$\hat{E}^{SL_{n}/P_{n-1,1}}(1, g;s)={\rm Res}_{\langle\lambda-\rho,\alpha_{i}^{\vee}\rangle=0,i=1,2,\ldots,n-2}\hat{E}^{SL_{n}/P_{1,\ldots,1}}(1, g;\lambda)$

denotes the simple roots of the root system $A_{n-1}$
associated to $SL_{n}$ , and
the Weyl vector.
is compact, andWith this, now notice that the moduli spacethat on the Levi of the Borel subgroup, 1 is cuspidal. So we can evaluate
the Eisenstein period

where$\alpha_{i}=\alpha_{i}-\alpha_{i+1}$$\rho=\frac{1}{2}\sum_{\alpha>0}\alpha$$\mathbb{M}_{\mathbb{Q},n}[1]$.

$\int_{SL_{n}(\mathbb{Z})\backslash SL_{n}(\mathbb{R})/O_{n}(\mathbb{R})}\Lambda^{0}\hat{E}^{SL_{n}/P_{1,\ldots,1}}(1, g;\lambda)$.

172This then gives a very precise expression of non-abelian zeta function
as a combination of terms consisting of products of rational functions coming from the symmetry depending only on the root system,
and abelian zeta functions. For details, please see [W]. As a direct consequence, we have the following$\hat{\zeta}_{\mathbb{Q},n}(s)$Fact 5. (Weng) (Parabolic Reduction, Stability & the Volumes)$m_{\mathbb{Q},n}^{ss}= \sum_{k\geq 1}(-1)^{k-1}\sum_{n_{1}+\cdots+n_{k}=n,n_{i}>0}\frac{1}{\prod_{j=1}^{k-1}(n_{j}+n_{j+1})}\cdot\prod_{j=1}^{k}m_{\mathbb{Q},n_{j}}.$Geometrically, this means that the semi-stable part can be obtained
from the fundamental domain associated toby deleting the tubular neighborhoods of cusps corresponding to parabolic subgroups whichparametrize the same type of canonical flags of unstable lattices, andwhose volumes, up to the lattice extensions, are completely determinedby that associated to the simple factors of related Levi factors. Undoubtedly, this parabolic reduction is also the ‘heart’ of the theory of
the truncations, both, analytic and arithmetic.
$SL_{n}$1.5

Parabolic Reduction: Geometric Theory

During our Sept, $2012$ ’s stay at IHES, Kontsevich introduced us theirbeautiful formula relating’s and. This basic relation is obtained within their lecture notes on the wall-crossing.
$m_{\mathbb{Q},n}$$m_{\mathbb{Q},n}^{ss\prime}s$

Fact 6. (Kontsevich-Soibelman) (Parabolic Reduction, Stability &the Volume)$\frac{1}{n}$

Here

.

$m \mathbb{Q},n=\sum_{k\geq 1}\sum_{n_{1}+\cdots+n_{k}=n,n_{i}>0}c_{n_{1},n_{2},\ldots,n_{k}}\cdot\prod_{j=1}^{k}m_{\mathbb{Q},n_{j}}^{ss}.$$c_{n_{1},n_{2},\ldots,n_{k}}$$:= \frac{1}{n_{1}(n_{1}+n_{2})\cdots(n_{1}+n_{2}+\cdots+n_{k})\cdots(n_{k-1}+n_{k})n_{k}}.$Indeed, the essence of this is the existence of the so-called canonicalfiltration, namely, the Harder-Narasimhan filtration of a lattice: For arank lattice , there exists a unique filtmtion of sub-lattices$n$$\Lambda$$0=\Lambda_{0}\subset\Lambda_{1}\subset\Lambda_{2}\subset\cdots\subset\Lambda_{k}=\lambda$

such that(i)$G_{i}(\Lambda);=\Lambda_{i}/\Lambda_{i+1}$

is semi-stable; and(ii) $Vol(G_{i}(\Lambda))^{rank(G_{i+1}(\Lambda))}\geq Vol(G_{i+1}(\Lambda))^{rank(G_{i}(\Lambda))}$

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2

Function$Fields/\mathbb{F}_{q}$The interactions between studies of number fields and that of function
fields (over finite fields) have been proven to be very fruitful, based onformal analogues between these two types of fields, despite the fact thatmany working mathematicians, not without their own reasons, believe
otherwise. The works presented here are yet another group of beautiful
examples.

2.1

Weil’s Formula: Tamagawa Numbers

Motivated by Siegel’s volume formula above, which originally was donein the theory of quadratic forms, Weil reinterpreted it in terms his famous
Tamagawa number one conjecture. For, this goes as follows.
Let $X$ be an irreducible, reduced, regular projective curve defined over
V. Denotes its function field by $F$ and its ring of adeles by . Fix avector bundleof rank on $X$ with determinant,Consider the groupwiththe maximal compact subgroup associated to $E$ . Then there is a natural morphism from thequotient spaceof rank
to the stackbundles with fixed determinant$SL_{n}$$\mathbb{A}$$E_{0}$$\lambda.$$n$$SL_{n}(\mathbb{A})$$\mathbb{K}(E_{0})$$\pi$$SL_{n}(F)\backslash SL_{n}(\mathbb{A})/\mathbb{K}(E_{0})$$n$$\mathbb{M}_{X,n}(\lambda)$$\lambda.$

Fact 7. The natural morphism
$\pi$

:

$SL_{n}(F)\backslash SL_{n}(\mathbb{A})/\mathbb{K}(E_{0})arrow \mathbb{M}_{X,n}(\lambda)$

is surjective with the fiberat the vector bundle
associated. Here $\det$ Aut
Aut
inunder the determinant mapping.

$\pi^{-1}(E_{0}^{g})$to $g\in SL_{n}(\mathbb{A})$ consisting ofdenotes the image of $\det$ Aut$E_{0}^{g}$$(E_{0}^{g}))$$\#(\mathbb{F}_{q}^{*}/\det$$(E_{0}^{g})$$(E_{0}^{g})$$\mathbb{F}_{q}^{*}$

Denote by
the moduli stack of rank bundle of degree on$X$ , and introduce
the total mass for rank and degree bundles on $X$
by$\mathbb{M}_{X,n}(d)$

$d$

$n$

$n$

$m_{X,n}(d):= \sum \frac{1}{\#Aut(E)}.$$E\in \mathbb{M}_{X,r\iota}(d)$Denote by$\hat{\zeta}_{X}(s):=\zeta_{X}(s)\cdot(q^{s})^{g-1}$

the complete Artin zeta function associated to $X$ , and$\hat{\zeta}_{X}(1):={\rm Res}_{s=1}\hat{\zeta}_{X}(s)\cdot\frac{l}{\log q}.$

$d$

174
Then Weil’s result that the Tamagara number of the quotient spaceequals one is equivalent to the following$SL_{n}(F)\backslash SL_{n}(A)$Fact 8. (Weil) (Tamagawa Number).

$mx_{n}(d)=mx_{n}:=\hat{\zeta}_{X}(1)\hat{\zeta}_{X}(2)\cdots\hat{\zeta}_{X}(n)$2.2Non-Abelian Zeta Functions for $X$

Denote by
the moduli stack of rank semi-stable bundle ofdegree on $X$ . Then define the pure non-abelian zeta function of rankfor $X$ by$n$$\mathbb{M}_{X,n}^{ss}(d)$

$d$

$n$

$\hat{\zeta}_{X,n}(s):=\sum \sum \frac{q^{h^{0}(X,E)}-1}{\#Aut(E)}\cdot(q^{-s})^{\chi(X,E)}.$$k\in \mathbb{Z}E\in \mathbb{M}_{X,n}^{ss}(kd)$Write$\hat{\zeta}_{X,n}(s)=\zeta_{X,n}(s)\cdot(q^{s})^{n(g-1)},$$Z_{X,n}(t):=\zeta_{X,n}(s)$with $t=q^{-s}$Introduce the partial mass of semi-stable bundles by$\alpha_{X,n}(d):=$$\sum$$\frac{q^{h^{0}(X,E)}-1}{\#Aut(E)},$$\beta_{X,n}(d):=$$\sum$$\frac{1}{\#Aut(E)}.$$E\in \mathbb{M}_{X,n}^{ss}(d) E\in \mathbb{M}_{X,n}^{ss}(d)$Then tautologically,$Z_{X,n}(t)= \sum_{m=0}^{(g-1)-1}\alpha_{X,n}(mn)\cdot(T^{m}+Q^{(g-1)-m}\cdot T^{2(g-1)-m})$$+ \alpha_{X,n}(n(g-1))\cdot T^{g-1}+(Q-1)\beta_{X,n}(0)\cdot\frac{T^{g}}{(1-T)(1-QT)}.$

where $T:=t^{n}$ and $Q:=q^{n}$ . This exposes the following
Fact 9. (Weng) (i) (Relation with Artin Zetas) $\zeta_{X,1}(s)=\zeta_{X}(s)$ ,the Artin zeta function for;(ii) (Rationality) There exists a degree $2g$ polynomial $P_{X,r}(T)\in \mathbb{Q}[T]$of such that
$X/\mathbb{F}_{q}$$T$$Z_{X,r}(t)= \frac{P_{X,r}(T)}{(1-T)(1-QT)}$(iii)$\beta$with$T=t^{r},$$Q=q^{r}$;unctional Equation)$\hat{\zeta}_{X,n}(1-s)=\hat{\zeta}_{X,n}(s)$;(iv) (Residues)$\hat{\zeta}_{X,n}(1):={\rm Res}_{s=1}\hat{\zeta}_{X,n}(s)\cdot\frac{1}{\log Q}=\beta_{X,n}(0)(=m_{X,n}^{ss}(0))$

.

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2.3

Parabolic Reduction, Stability and the Mass: Geometric Theory

To go further, make a normalization by introducing$\tilde{m}_{X,n}^{ss}(d)=\frac{1}{q^{\frac{n(n-1)}{2}(g-1)}}$.

$m_{X,n}^{ss}(d)$.

Then the parabolic reduction via the Harder-Narasimhan filtration leads
to the following relation involving infinite sums:
Fact 10. (Harder-Namsimhan, Desale-Ramanan) (Parabolic Reduction).
$m_{X,n}(d)= \sum_{k\geq 1}\sum_{n_{1}+\cdots+n_{k}=n,n_{i}>0\lrcorner^{d}}\sum_{+d_{1+}^{n_{1}}\cdot\cdot d_{k}d}q^{-\Sigma_{i<j}(d_{i}n_{j}-d_{j}n_{i})}\prod_{j>,.\cdots>\frac{d_{k}}{n_{k=}}=1}^{k}\tilde{m}_{X,n_{j}}^{ss}(d_{j})$2.4Parabolic Reduction, Stability and the Mass: Combinatorial Aspect

With the above result, Zagier proved the following fundamental result,hidden in his paper on Verlinder formula:
Fact 11. (Zagier) (Parabolic Reduction, Stability & the Mass)$\tilde{m}_{X,n}^{ss}(d)=\sum_{k\geq 1}(-1)^{k-1}\sum_{n_{1}+\cdots+n_{k}.=nn_{i}>0,i=1,k}..,\prod_{j=1}^{k-1}\frac{q^{(n_{j}+n_{j+1})\cdot\{(n_{1}+\cdots+n_{j})\cdot\frac{d}{n}\}}}{q^{(n_{j}+n_{j}+1)}-1}\cdot\prod_{j=1}^{k}m_{X,n_{j}}.$This formula should be compared with with our Fact 5 for number
fields. The structure are very much similar: in fact if we let $qarrow 1$ , thenwe would get the number theoretic identity there. Indeed, the original
formula of Zagier is a bit different: original coefficients depends on
Motivated by our number fields analogue, we reorganize it with for $m$. Consequently, our coefficients are environmentally free.
andforThat is, independent of the curve $X$ and the genus.
$g.$$\tilde{m}$$\hat{\zeta}_{X}$2.5$\zeta_{X}$Parabolic Reduction, Stability and the Mass: New
FormulaThe above relation of Harder-Narasimhan, Ramanan-Desale and Zagier for function fields correspond to our own formula listed as Fact 5.

176
So naturally, what should be the one appeared in the theory of parabolic
reduction, stability and the volumes obtained by Kontsevich-Soibelman.

This was recently obtained by Zagier during his visit to Fukuoka in May,

2012.
Fact 12. (Zagier) For an ordered partition $n=n_{1}+n_{2}+\cdots+n_{k}$ , fix$\delta_{i}\in\{0, , \ldots, n_{1}-1\}$ , thensatisfying
fix$v_{i}\in[0,1)\cap \mathbb{Q}$$v_{i} \equiv\frac{\delta_{i}}{n_{i}}-\frac{\delta_{i+1}}{n_{i+1}}(mod 1)$.

Also set $N_{i}=n_{1}+n_{2}+\cdots+n_{i}$ and $N_{i}’=n-N_{i}$ for $i=1,2,$$\ldots n.$(i) (On Average)$n \cdot m_{X,n}=\sum_{1}\sum_{n_{i}>0,i=12,..,k},\sum_{j=1,2,\ldots,k},\prod_{ik\geq n_{1}+n_{2}+\cdots+n_{k},.=n\delta_{i}\in\{0,1,\ldots n_{i}-1\}=1}^{k-1}\frac{q^{v_{i}N_{i}N_{i}’}}{q^{N_{i}N_{i}’}-1}\cdot\prod_{j=1}^{k}\tilde{m}_{X,n_{j}}^{ss}(\delta_{j})$(ii) ([ndividuality) For all $d=0,1,$$\ldots,$.

$n-1,$$m_{X,n}= \sum_{k\geq 1}\sum_{n_{1}+n_{2}+\cdots+n_{k}.=nn_{i}>0,i=12,..,k},\frac{1}{n}$$\cross\sum_{j=1,2,\ldots,k}\delta_{i}\in\{0,1,\ldots,n_{i}-1\}(\sum_{\zeta_{n}\cdot\zeta_{n}^{n}=1}\zeta_{n}^{n-d}\cdot\prod_{h=1}^{k-1}\frac{\zeta_{n^{h}}^{vN_{h}}q^{v_{h}N_{h}N_{h}’}}{\zeta_{n}^{N_{h}}q^{N_{h}N_{h}’}-1})\prod_{j=1}^{k}\tilde{m}_{X,n}^{ss}j(\delta_{j})$The first is obtained by taking average on from the relation of Fact
11, while the second is obtained directly from that of Fact 11. With
geometric picture in mind, formula (ii) should be further polished, so as
to get everything done according to the real world structure. This may
prove to be a bit complicated due to the fact that usually$d$$\mathbb{M}_{X,n}(d)\not\simeq \mathbb{M}_{X,n}(d’) , \forall d, d’\in\{0,1, \ldots, n-1\}, d\neq d’.$This is very different from the cases for number fields, where we always
have the isomorphism $(*)$ between different levels.
We remind the reader that while all relations in function field case are
obtained using geometric methods, our basic relation for number fields
are obtained analytically using Eisenstein series.

.

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3

Parabolic Reduction, Stability and the Mass:
General Reductive Groups

Motivated by the above discussion, more generally, for a split reductive
Borel, , denote bygroup defined over a number field $F,$a
the adelic elements of corresponding to semi-stable principle Glattices ([G]). Writefor the associated maximal compact subgroup.
Also for a standard parabolic subgroup , write its Levi decomposition
as $P=UM$ with the unipotent radical and $M$ its Levi factor. Denotethe corresponding simple decomposition of $M$ as
with’s thesimple factors of $M$ . Introduce invariants
$G$

$B$

$fi\grave{x}ed$$\cdots$$G$$G(\mathbb{A})^{ss}$$\mathbb{K}_{G}$

$P$

$U$

$M_{i}$$\prod_{i}M_{i}$$m_{F;P}:= \prod_{i}Vol(\mathbb{K}_{M_{i}}\backslash M_{i}^{1}(\mathbb{A})/M_{i}(F)Z_{M_{i}^{1}(\mathbb{A})})$

and$m_{F;P}^{ss}:= \prod_{i}Vol(\mathbb{K}_{M_{i}}\backslash M_{i}^{1}(\mathbb{A})^{ss}/M_{i}(F)Z_{M_{i}^{1}(A)})$.

In parallel, we have similar constructions for function fields $F=\mathbb{F}_{q}(X)$ .
Denote by$n_{i}:=\#\{\alpha>0:\langle\rho, \alpha^{\vee}\rangle=i\}-\#\{\alpha>0:\langle\rho, \alpha^{\vee}\rangle=i+1\}$

and by$v_{G}$

the volume of$\{\sum_{\alpha\in\triangle} :a_{\alpha}\alpha^{\vee}:a_{\alpha}\in[0,1)\}.$

Fact 13. (Langlands) (Volume of Fundamental Domain) For thefield of mtionals,$Vo1(\mathbb{K}_{G}\backslash G^{1}(\mathbb{A})/G(\mathbb{Q})Z_{G^{1}(\mathbb{A})})=vG^{\cdot}\prod_{i\geq 1}\hat{\zeta}(i)^{-n}i.$Based on all this, then we have the following
Conjecture 1. (Weng) (Parabolic Reduction) Let $G/F$ be a split reductive group with $B/F$ a fixed Borel. Then, for each standard parabolic, independent
subgroup $P$ of , there exist constantsof $F$ , such that$G$

$m_{F;}c= \sum_{P}C_{P}\cdot m_{F;P}^{SS},$$c_{P}\in \mathbb{Q},$$m_{F;G}^{ss}= \sum_{P}$$e_{P}\in \mathbb{Q}_{>0}$sgn$(P)\cdot e_{P}\cdot m_{F;P},$

where $P$ runs over all standard pambolic subgroups ofdenotes the sign of $P.$$G$

, and sgn $(P)$178
The exact values of$e_{P}$’s can be written out in terms of the associatedroot system. Indeed, if$W_{0}:=\{w\in W:\{\alpha\in\triangle:w\alpha\in\triangle\cup\Phi^{-}\}=\triangle\},$and thethen there is a natural one-to-one correspondence between
set of subsets of , and hence to the set of standard parabolic subgroupsof . Thus we will write$W_{0}$$\triangle$$G$$W_{0}:=\{wp;P$ standard parabolic subgroup

and, for $w=Wp\in W_{0}$ , write$J_{P}\subset\Delta$$\},$

the corresponding subset.

Conjecture 2. (Weng) Let be a split type reductive group with $P$ its
maximal pambolic subgroup.
(1) For a number field $F,$(i) (Relation to Zetas with Symmetries)$G$$m_{F;G}^{ss}={\rm Res}_{s=-c_{P}}\hat{\zeta}_{F}^{\langle G,P)}(s)={\rm Res}_{\lambda=\rho}\omega_{F}^{G}(\lambda)$;(ii) (Parabolic Reduction, Stability & the Mass)$m_{F;G}^{ss}= \sum_{P}\frac{(-1)^{rank(P)}}{\prod_{\alpha\in\Delta\backslash w_{J}J_{P}}(1-\langle wJ\rho,\alpha^{\vee}\rangle)}$. $mF;P$ ;(2) For an irreducible reduced regular projective curve $X,$(i) (Relation to Zetas with Symmetries);$\log q\cdot m_{F;G}^{ss}={\rm Res}_{s=-c_{P}}\hat{\zeta}_{X}^{(G,P)}(s)={\rm Res}_{\lambda=\rho}\omega_{X}^{G}(\lambda)$(ii) (Parabolic Reduction, Stability & the Mass)$m_{F;G}^{ss}= \sum_{P}\frac{(-1)^{rank(P)}}{\prod_{\alpha\in\triangle\backslashw_{J}J_{P}}(1-q^{\langle w_{J}\rho,\alpha^{\vee}\rangle-1})}\cdot mF;P.$

Remark. Calculations in [Ad] for lower rank groups indicate that, for. It would be very interesting to find a close
number fields,
formula for them.
$\frac{1}{c_{P}}\in \mathbb{Z}_{>0}$

Lin WENG, Institute for Fundamental Research, The Academy andGraduate School of Mathematics, Kyushu University, Fukuoka, 8190395, JAPAN-Mail: weng@math.kyushu-u.ac.jp
$L$

$E$

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REFERENCES[Ad] K. Adachi, Parabolic Reduction, Stability and Volumes of Fundamental Domains: Examples of Low Ranks, Master Thesis, Kyushu
Univ., 2012
[Al] J. Arthur,
to classes in$A$trace formula for reductive groups. I. Terms associated. Duke Math. . 45 (1978), no. 4, 911-952$G(\mathbb{Q})$$J$[A2] J. Arthur,
trace formula for reductive groupsII. Applications ofa truncation operator. Compositio Math. 40 (1980), no. 1, 87-121.
$A$[DR] U.V. Desale & S. Ramanan, Poincare polynomials of the variety ofstable bundles, Math. Ann 26 (1975) 233-244[G] D. R. Grayson, Reduction theory using semistability. Comment.
Math. Helv. 59 (1984), no. 4, 600-634.
[HN] G. Harder& M.S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212, 215-248 (1975)[K] M. Kontsevich-Soibelman, Lecture Notes on Wall-crossing, Dec 2011
[L] L. Lafforgue, Chtoucas de Drinfeld et conjecture de RamanujanPetersson. Asterisque No. 243 (1997)[La] R. Langlands, The volume of the fundamental domain for somearithmetical subgroups of Chevalley groups, Proc. Sympos. Pure Math.
9, AMS (1966) pp.143-148[Wl] L. Weng, Non-abelian zeta functions for function fields. Amer.
Math. 127 (2005), no. 5, 973-1017.

$J.$[W2] L. Weng,geometric approach to -functions. The Conference
on -Functions, 219-370, World Sci. Publ., Hackensack, $NJ$ , 2007.
$A$

$L$

$L$

[W3] L. Weng, Symmetries and the Riemann Hypothesis, Advanced
Studies in Pure Mathematics 58, Japan Math. Soc., 173-224, 2010

[W4] L. Weng, Counting Bundles, arXiv:1202.0869[W5] L. Weng, Zeta Functions for Elliptic Curves I: Counting Bundles,
arXiv: 1202.0870[W6] L. Weng, Parabolic Reduction, Stability and The Mass, in preparation[Zl] D. Zagier, Elementar\’y aspects of the Verlinde formula and theHarder-Narasimhan-Atiyah-Bott formula, in Proceedings of the Hirzebruch 65 Conference on Algebmic Geometry, 445-462 (1996)[Z2] D. Zagier, private notes, May, 2012