Introduction
Gauss
Gauss Gauss Euler
( ∂ := dx) Gauss (GE): Pu = 0 ** 7
1

x λ xP = x²∂² + γx∂ − x(x²∂² + (α + β + 1)x∂ + αβ) = ϑ(ϑ − 1) + γϑ − x(ϑ(ϑ − 1) + (α + β + 1)ϑ + αβ) = ϑ(ϑ + γ − 1) − x(ϑ + α)(ϑ + β). ∴ Ри = 0 ⇐⇒ ϑ(ϑ + γ − 1)u = x(ϑ + α)(ϑ + β)u. u(x) = ∑ π=0 cnxn x n
Gauss 2

2
xn
F(α,β,γ; x)
x = 0 φι (0) = φ 2 (0) = 1 Φ1(x), Φ2(x) u(x) = C 1 x^o,1 φ 1 (x) + C 2 x^o,2 φ 2 (x) (λ 0,1 = 0, λ 0,2 = 1 − γ, C 1, C 2 ∈ C) 0 x = 0 (characteristic exponent) φι ( x ) = F ( α, β, γ; x )
{α,β}
Riemann (Riemann scheme) Gauss Gauss Riemann *8 (monodromy) (CU{∞})\{0,1,∞} Xo (u1, u2) xo Yj (j = 0, 1, ∞) XO M j ∈ GL(2, C) Y j (u1, u2) Yj(u1,u2) = (u1,u2)Mj
*6 1
×7 *8 rigid

Kac-Moody root
2
u e(λ02)e(λ11) + e(λ01)e(λ12) + ab = e(−λ21)e(−λ22) Riemann rigid Riemann Gauss Fuchs Deligne-Simpson-Katz Fuchs X x = 1 *9 ×10 b 1. X U U
n
F
2.
0
C, N, e
3. n
Wronskian X
. F P Pu = 0
Fuchs 2.1
u1(x),...,un(x)
cj
5
F M j
Cj
n = m j,1 + + · · + m j,n j {\j,v + k ; k = 0,...,mj,v − 1, v = 1,...,nj} λj,ν − λj,ν ≠ Z  (1 ≤ ν < ν ′ ≤ nj)
(7)
λj,ν − λj,ν ′̸≠ Z \ {0} (1 ≤ ν ≤ ν ′ ≤ n j ) m j,1 ≥ m j,2 ≥ ··· ≥ m j,n j (6) c; M j rank ( M j − e 2π√−1λ j,1 ) . . . ( M j − e 2π√−1λ j,k ) = n − m j,1 − · · · − m j,k

6
[Os4]
Riemann scheme

F
Jordan-Pochhammer Riemann scheme
n = ord m
Fuchs 2. {\m}
Riemann scheme (GRS) Fuchs (FC)

7
m
Fuchs

Fuchs m irreducibly   realizable Fuchs
2.2
Kac-Moody root

W ∞ Kac-Moody root

8
w ∈ W∞, α ∈ Q

2.3
reduction

addition
9

Weyl
Euler
m' = ∂ε m, GRS generic
∂ę P Fuchs ord ℓ m = ord m − d ℓ (m). 0 m j,ℓ j ord ∂ ε m ∂ ℓ m l l max (m) M j,ℓ max(m) j = max{ m j,1,..., m j,n j } (26) Amax P dmax (m), ∂max { λm } θmax m = θε max (m) K irreducibly realizable reduction m

Weyl
10
ord m > ord ∂max m > ord ∂ max m > · · · > ord ∂max m, ord K m = 1  or  dmax(∂ max m) ≤ 0. l ×12 C 0 {\m} Kac-Moody
Schlesinger
{ Pm：Fuchs ↓ ∂ε , addition
3 (
{^m} λj,v generic GRS Fuchs { Pm：Fuchs 91, ・・・, 9N
irreducibly realizable m idx m = 2 rigid 1 indivisible m {m j,v} basic indivisible monotone dmax(m) ≤ 0 m idx m m j,3 ≥ ··· ¯ D 4 ,  Ë 6 ,  Ë 7 ,  Ë 8 affine root idx m  = 0 13 idxm = −2 [Os2] m irreducible realizable reduction (27) mj,1 ≥ mj,2 ≥ (cf. [Os2]). 4 cf. [Ko] ord ∂ max m = ord ∂ i − 1 simply reducible idx m rigid non-rigid m (cf. [Os4]) 2111, 222, 33 Simpson in [Si] m simply  reducible simply reducible (cf. [MWZ]) 10269 2.4 reduction irreducible realizable Fuchs (30) 306 rigid [Os4] or m (32)

13

14
Gauss Gauss 
 

3.1
Riemann
1
P ∈ W [ x ]
8.
n
u v,i (x) — ef λv,(x-1)夢∈ xmv ef λv,(x-1)夢Ô   (0 ≤ i ≤ mv, — 1, ν = 1,..., N) Uv,i
(41) ef λν(x-1) ☞ := x λν,o e
l j,v,v' lj,v,v,v = Z=0, lj,v,v = 0, lj,v,v = lj,v,v and l ≥יע,v,j, = 0 ≤ j ≤ p, 1 ≤ v, v, v, v, ≤ n j )
2.1.
T j n
r j
212|32|32,41 = ((21))((2)),41
Riemann
Reduction (cf. §3.3) : 212|32|32,41  →  12|12|12,21 → 1|1|1,1 = 1,1 3.2 Fuchs Fourier-Laplace transform:
L :  W[x] → W[x], x → −∂, ∂ → x Reduced form (R(P)): , R(P) = ∑ pi(x) ∂i ∈ C(x)× P ∩ W [x] C[x] P ∈ W ( x ) p i (x) 1 monic
Addition   Gauge

16
ef (x)
x Euler Twisted Euler transform (middle convolution): P ∈ W ( x ) μ ∈ C
3.3
Kac-Moody root

17

13 (Versal Gauss : 11|11|11). Ì = (1 − c 1x )(1 − c 2x ) ∂ 2 +((λ 2 + 2c 1c 2 )x + λ 1 − c 1 − c 2 ) ∂ +μ(λ 2 + c 1c 2 (1 − μ)) = (1 − c 1x )(1 − c 2x ) ∂² +( ζ 2x + ζ 1 ) ∂ + μ ( ζ 2 − c 1c 2 ( μ + 1)) (c 1 = c 2 = 0)

3.4
Kac-Moody

N i,0 = 0

19

20
Weyl Q L Q L = {m}) (m, l) W Φ(α) 1. Φ 2. Weyl Kac-Moody歌Fuchs 15 ([HO]). ∀m ∈ L+ Φ−¹(m) ∩ Q ξ (:= ⊕αες Z≥oα) ≠ ∅. generic Kac-Moody Φ−1(m)∩Ω± 3.5 idx P = 2 rigid twisted Euler transformations additions (m, l) l d(ℓ) > 0 basic m m = ( m j v ) L P (m, l) 16 ([HO]). index of rigidity r basic r = 0 r < 0 basic idx P = 0, −2 Fuchs 17.  P Fuchs additions i) ([Ko]) idx P = 0 twisted Euler transformations 4 P (11, 11, 11),  (111, 111, 111),  (22, 1111, 1111),  (33, 222, 111111) ii) ([Os2]) idx P = −2 twisted Euler transformations additions 13 P (11, 11, 11, 11, 11), (21, 21, 111, 111),     (31, 22, 22, 1111),     (22, 22, 22, 211), (211, 1111, 1111), (221, 221, 1111),     (32, 1111, 1111),     (222, 222, 2211), (44, 2222, 22211),     (44, 332, 1111111),     (55, 3331, 22222), (33, 2211, 11111), (66, 444, 222211) 18.  P P i)  idx P  =  0 additions (m, l) P min twisted Euler transformations歌( mmin , l) Dynkin α(Pmin) ×14 support E(1), E(1), D(1), A(1), A(1), A(1) ⊕ A(1). twisted Euler transformations ii) ([HO]) idx P = −2 additions歌P min (mmin, l) Dynkin support α(Pmin) α( P min ) P min Lmin C L(P) Φ | Lmin 1 Q "Sタニρ ♀ 2グタ2 ρ ♀† 2 2 01 ♀† 1 1
Here a, b ∈ Z. 2 C 1 (
タ2 pl
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