Fractional non-Archimedean differentiability [Elektronische Ressource] / vorgelegt von Enno Nagel

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• Mathematics

Informations

• Publié par: westfalische_wilhelms-universitat_munster
• Publié le: 01 janvier 2011
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Fractional Non-Archimedean
Differentiability
Enno Nagel
2011Mathematik
Fractional Non-Archimedean
Differentiability
Inaugural-Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften im Fachbereich
Mathematik und Informatik
der Mathematisch-Naturwissenschaftlichen Fakultät
der Westfälischen Wilhelms-Universität Münster
vorgelegt von
ENNO NAGEL
aus Stadtlohn
-2011-Dekan: PROFESSOR DR. MATTHIAS LÖWE
Erster Gutachter: PROFESSOR DR. PETER SCHNEIDER
Zweiter PROFESSOR DR. CHRISTOPHER DENINGER
Tag der mündlichen Prüfung: 01.02.2011
Tag der Promotion:Abstract
Letr≥ 0 be real number letK be a complete non-Archimedeanly non-trivially valued field.
dIn the first chapter, we give the definition of a functionf : X→ E on a domainX⊆ K
rwith values in aK-Banach spaceE to ber-times differentiable orC at a pointa∈X. Then
rwe endow the K-vector space of all suchC -functions with a locally convex topology and
examine properties of theirs such as completeness, density of (locally) polynomial functions,
closure under composition and, for the dual, under convolution.
For functions on open domains in one variable, we show this definition to equal a handier de-
r
scription through the convergence speedo(1/|h| ) of the rest term of the Taylor-polynomial at
dx +h expanded aroundx up to degreebrc. Moreover on the special domainX =Z we showp
r dtheC -functionsf :Z →K to be characterized by its Mahler coefficients (a ) d obeyingn n∈Np
r
|a ||n| → 0 as|n|→∞, where we put|n| :=n +···+n . Then as a corollary, a characteri-n 1 d
r dzation ofC -functionsf :X→K on openX⊆Q by partial Taylor-polynomials is obtained.p
We turn to the second chapter: LetG be a connected reductive group over a local fieldF andP
a minimal parabolic subgroup. LetK be a complete non-Archimedeanly non-trivally valued
∗field of characteristic 0 with valuation ringo. Letθ :P→K be an unramified character and
G
denote byI(θ) = Ind θ the smooth principal series. LetU be an algebraic representation ofP
G (and ifU is nontrivial, also assumeK⊇F andG to split). ThenV =I(θ)⊗ U is a locallyK
balgebraicG-representation, and we letV be the universalK-Banach space with
aG-invariant norm whereintoV maps continuously with respect to its finest locally convex
topology. We will then show that the universal unitary lattice L⊆ V , given by the preimage
P
bof the unit ball inV , is of the formL = L withW denoting the Weyl group ofG andw∈W w
¯eachL being a cyclico[P ]-module which is free as ano-module.wAcknowledgements
First of all, I would like to thank very much all my dear former and current colleagues of my
work group, and in particular my advisor Peter Schneider.
I also greatly profited from exchange with Stany de Smedt, Helge Glöckner, Wim Schikhof
and Benjamin Schraen. I am moreover indebted to Marco de Ieso for careful correction read-
ing. It is a pleasure to thank everyone of them.
Most importantly though, I want to thank my family for its support.Contents
Fractional non-Archimedean calculus 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 Locally convexK-vector spaces . . . . . . . . . . . . . . . . . . . . 6
ρ1.2 C -functions forρ∈ [0, 1[ . . . . . . . . . . . . . . . . . . . . . . . 7
ρDefinition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 7
ρProperties of the space ofC -functions . . . . . . . . . . . . . . . . . 9
ρThe locally convex topology onC -functions . . . . . . . . . . . . . 11
ρComponentwise criteria for beingC . . . . . . . . . . . . . . . . . . 15
ρρρ d1.3 C -functions forρρρ∈ [0, 1[ . . . . . . . . . . . . . . . . . . . . . . . 18
ρρρDefinition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 19
ρProperties of the space ofC -functions . . . . . . . . . . . . . . . . . 20
1+ρ1.4 C -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Fractional differentiability in one variable . . . . . . . . . . . . . . . . . . . 32
r2.1 C -functions forr∈R . . . . . . . . . . . . . . . . . . . . . . . . 32≥0
rDefinition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 32
rProperties ofC . . . . . . . . . . . . . . . . . . . . . . . 33
rThe locally convexK-algebra ofC -functions . . . . . . . . . . . . . 35
Description through iterated difference quotients . . . . . . . . . . . 38
2.2 Characterization through Taylor polynomials . . . . . . . . . . . . . 39
rThe Taylor polynomial ofC -functions . . . . . . . . . . . . . . . . 39
rCharacterizingC -functions through Taylor polynomials on general
domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
rSufficiency of the Taylor polynomial expansion on B -sets forC -ν
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
rAnother characterization ofC -functions on compact sets and an ap-
plication . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Orthogonal bases onZ . . . . . . . . . . . . . . . . . . . . . . . . 57p
ρThe van der Put base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . 58p
ρThe Mahler base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 59p
rThe Base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 65p
3 Fractional differentiability in many variables . . . . . . . . . . . . . . . . . . 71
r3.1 C -functions forr∈R . . . . . . . . . . . . . . . . . . . . . . . . 71≥0
rDefinition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 71
rProperties ofC . . . . . . . . . . . . . . . . . . . . . . . 74rThe locally convexK-algebra ofC -functions . . . . . . . . . . . . . 77
rLocally analytic functions inC (X,K) on an open domain . . . . . . 82
rComposition properties ofC -functions . . . . . . . . . . . . . . . . 85
rDensity of (locally) polynomial functions inC (X,K) . . . . . . . . 90
r3.2 Orthogonal bases ofC -functions on a compact domain . . . . . . . . 98
Interlude: Orthogonal bases ofK-Banach spaces . . . . . . . . . . . 98
rThe initialK-Banach algebraC (X,K) of thought topological tensor
rrr dproductsC (X,K) forrrr∈N . . . . . . . . . . . . . . . 100=r
r dThe Mahler base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 106p
r d3.3 Description ofC (X,K) for openX⊆Q through Taylor polynomials 108p
r r3.4 The spaceD (X,K) of distributions onC (X,E) for a compact group
X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
rExample of an inducedC -representation . . . . . . . . . . . . . . . 124
rC -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
The intertwined open cells in the universal unitary lattice of an unramified
algebraic principal series 133
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
0 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
0.1 The groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
0.2 The representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
0.3 The universal unitary completion of a locally algebraic representation 137
1 The unramified dominant principal series as a representation ofP . . . . . . 140
2 The universal unitary lattice of the P -representation on an open cell and a
norm of differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . 144
Interlude: The dominant submonoid acting on the affine root factors . . . . . 144
2.1 The necessity criterion . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.2 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Example: The smooth case of small order image . . . . . . . . . . . 155 The split case . . . . . . . . . . . . . . . . . . . . 156
Interlude: Locally polynomial differentiable functions . . . . . . . . . . . . . 157
2.3 The locally algebraic case . . . . . . . . . . . . . . . . . . . . . . . 164
3 The universal unitary lattice of an unramified dominant principal series . . . . 168
3.1 The universal unitary lattice of the underlyingP -representation . . . 168
3.2 The Jacquet module . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.3 Gluing the universal unitary lattice from the intertwined open cells . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Fractional non-Archimedean calculus
Introduction
Start with two normed finite dimensional vector spacesV andW over a valued fieldK. Let
f : U→ W some map defined on an open subsetU⊆ V . Thenf is called differentiable or
1C in the pointa∈U if there exists a linear mapD :V→W such that for everyε> 0 therea
is a neighborhoodU 3a inU withε
kf(x +h)−f(x)−D ·hk≤εkhk for allx +h,x∈U .a ε
To iterate this differentiability notion, we need a choice of coordinates on the function’s do-
dmain. We therefore assumeV = K and lete ,...,e be its canonical basis vectors. Then1 d
d∗ ]1[given any two points x +h,x∈ U with h∈ K , we can define A := f (x +h,h)∈
Hom (V,W ) by the partial difference quotientsK-vctsp.
A(h ·e ) =f(x