Symbolic calculus on coadjoint orbits of nilpotent Lie groups - - PowerPoint PPT Presentation

▶
symbolic calculus on coadjoint orbits of nilpotent lie
SMART_READER_LITE
LIVE PREVIEW

Symbolic calculus on coadjoint orbits of nilpotent Lie groups - - PowerPoint PPT Presentation

Oct 26, 2023 234 likes •406 views

Symbolic calculus on coadjoint orbits of nilpotent Lie groups Daniel Beltit a Institute of Mathematics of the Romanian Academy *** Joint work with Ingrid Beltit a (IMAR) Bia lowie za, June 27, 2012 Daniel Beltit a

slide-1
SLIDE 1

Symbolic calculus on coadjoint orbits

  • f nilpotent Lie groups

Daniel Beltit ¸˘ a Institute of Mathematics of the Romanian Academy *** Joint work with Ingrid Beltit ¸˘ a (IMAR) Bia lowie˙ za, June 27, 2012

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 1 / 16

slide-2
SLIDE 2

References

◮ I. Beltit

¸˘ a, D. B., Boundedness for Weyl-Pedersen calculus on flat coadjoint orbits. Preprint arXiv:1203.0974v1 [math.AP].

◮ I. Beltit

¸˘ a, D. B., Modulation spaces of symbols for representations of nilpotent Lie groups. J. Fourier Analysis Appl. 17 (2011), no. 2, 290–319.

◮ I. Beltit

¸˘ a, D. B., Algebras of symbols associated with the Weyl calculus for Lie group representations. Monatshefte f¨ ur Math. (to appear).

◮ I. Beltit

¸˘ a, D. B., Continuity of magnetic Weyl calculus. J. Functional Analysis 260 (2011), no. 7, 1944-1968.

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 2 / 16

slide-3
SLIDE 3

Plan

1 Early origins of the pseudo-differential Weyl calculus:

  • H. Weyl (1928), M.E. Taylor (1968), R.F.V. Anderson (1969),
  • L. H¨
  • rmander (1979)

2 Orbital Weyl calculus for unirreps of nilpotent Lie groups:

N.V. Pedersen (1994)

3 Continuity of pseudo-differential operators Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 3 / 16

slide-4
SLIDE 4

Origins of the pseudo-differential Weyl calculus

Quantum positions Q1, . . . , Qn and momenta P1, . . . , Pn (Qjf )(q) = qjf (q) and Pjf = 1 i ∂f ∂qj for j = 1, . . . , n. Fact: for p = (p1, . . . , pn) ∈ Rn and q = (q1, . . . , qn) ∈ Rn, q · Q + p · P := q1Q1 + · · · + qnQn + p1P1 + · · · + pnPn is a self-adjoint oper. in L2(Rn), so there exists the unitary ei(q·Q+p·P). Weyl-H¨

  • rmander calculus a ❀ a(Q, P) (symbol ❀ pseudo-diff. oper.)

a(Q, P) =

  • Rn×Rn

ei(q·Q+p·P)

  • π(q,p,0)
  • a(q, p) dpdq

: S(Rn) → S′(Rn) “Integral” kernel formula: (a(Q, P)f )(q) =

  • Rn×Rn

a q + q′ 2 , p

  • ei(q−q′)·pf (q′)dp dq′

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 4 / 16

slide-5
SLIDE 5

Schr¨

  • dinger repres. of the Heisenberg group

Canonical commutation relations

[Qj, Qk] = [Pj, Pk] = 0 and [Qj, Pk] = δjki · I for j, k = 1, . . . , n

  • Heisenberg algebra: h2n+1 = Rn × Rn × R

[(q, p, t), (q′, p′, t′)] = [(0, 0, p · q′ − p′ · q

  • symplectic form on Rn×Rn

)]

  • Heisenberg group: H2n+1 = (h2n+1, ∗),

X ∗ Y := log(eXeY ) = X + Y + 1 2[X, Y ]; unit element 0 ∈ H2n+1, inversion X −1 := −X.

  • Schr¨
  • dinger representation π: H2n+1 → B(L2(Rn)),

π(q, p, t)f (x) = ei(px+ 1

2 pq+t)f (q + x) Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 5 / 16

slide-6
SLIDE 6

Orbital Weyl calculus for nilpotent Lie groups, pag. 1 (classical phase space)

  • G nilpotent Lie group with Lie alg. g and pairing ·, ·: g∗ × g → R

hence [g, [g, . . . [g, g] . . . ]] = {0}; e.g., [h2n+1, [h2n+1, h2n+1]] = {0}.

  • coadjoint orbit O ֒

→ g∗

  • ξ0 ∈ O ❀ gξ0 = {X ∈ g | ξ0 ◦ adgX = 0}
  • {0} = g0 ⊂ g1 ⊂ · · · ⊂ gm = g Jordan-H¨
  • lder sequence

❀ the set of jump indices e ⊂ {1, . . . , n} for O

  • Xj ∈ gj \ gj−1 for j = 1, . . . , m

❀ ge = span{Xj | j ∈ e} ⊆ g ❀ O ∼ → g∗

e, ξ → ξ|ge

  • Fourier tr. a → ˆ

a =

  • O e−iξ,•a(ξ)dξ

S(O) ∼ → S(ge), L2(O) ∼ → L2(ge)

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 6 / 16

slide-7
SLIDE 7

Orbital Weyl calculus for nilpotent Lie groups, pag. 2 (quantization of the phase space)

  • π: G → B(H) unirrep assoc. with O
  • H∞ = {φ ∈ H | π(·)φ ∈ C∞(G, H)} space of smooth vectors

H−∞ (=anti-dual of H∞) sp. of distribution vectors ❀ H∞ ֒ → H ֒ → H−∞ (e.g., S(Rn) ֒ → L2(Rn) ֒ → S′(Rn))

  • π⊗2 : G × G → B(S2(H)), π⊗2(g1, g2)T = π(g1)Tπ(g2)−1

❀ B(H)∞ = {T ∈ S2(H) | π⊗2(·)T ∈ C∞(G × G, S2(H))}

  • ≃L(H−∞,H∞) regularizing opers.

֒ → S1(H)

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 7 / 16

slide-8
SLIDE 8

Orbital Weyl calculus for nilpotent Lie groups, pag. 3 (quantization of the observables)

g∗ ⊃ O ← → π: G → B(H)

Weyl-Pedersen calculus (for a ∈ S′(O))

Opπ(a) =

  • ge π(expG X)ˆ

a(X)dX : H∞ → H−∞

Properties (N.V. Pedersen, 1994)

◮ Opπ: S(O) ∼

→ B(H)∞, S′(O) ∼ → L(H∞, H−∞), L2(O) ∼ → S2(H)

◮ Opπ(·, X|O) = dπ(X) for X ∈ g ◮ Tr(Opπ(a)) =

  • O a(ξ)dξ

◮ Opπ(¯

a) = Opπ(a)∗

Earlier results: A. Melin (1981–1983), K.G. Miller (1982), D. Manchon (1991)

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 8 / 16

slide-9
SLIDE 9

Analysis of the square-integrable repres., pag. 1

  • Z the center of the group G
  • repres. π square integrable mod. Z with the coadjoint orbit O

Definition: (∃ψ ∈ H \ {0})

  • G/Z |(π(·)ψ | ψ)|2 < ∞

Characterization: O is an affine subspace in g∗ Example: The orbit of the Schr¨

  • dinger repres. is R2n × {1} ֒

→ R2n+1.

Theorem (covariance of the Weyl-Pedersen calculus)

a ∈ S′(O), g ∈ G ⇒ Op(a ◦ Ad∗

G(g)|O) = π(g)−1Op(a)π(g)

Diff (O) = the set of all linear differential operators D : C∞(O) → C∞(O) which are invariant to the coadjoint action: (∀g ∈ G)(∀a ∈ C ∞(O)) D(a ◦ Ad∗

G(g)|O) = (Da) ◦ Ad∗ G(g)|O.

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 9 / 16

slide-10
SLIDE 10

Analysis of the square-integrable repres., pag. 2

Sp(H) =      all compact operators if p = 0, {T ∈ B(H) | trace |T|p < ∞} if 1 ≤ p < ∞, B(H) if p = ∞.

Main Theorem

Assume dim O = dim G − dim Z and p ∈ {0} ∪ [1, ∞].

1 Equivalent assertions for a ∈ C ∞(O): ◮ for every D ∈ Diff (O) we have Da ∈ Lp(O), not.: a ∈ C∞,p(O); ◮ for every D ∈ Diff (O) we have Op(Da) ∈ Sp(H). 2 C∞,p(O) has a Fr´

echet topology defined by any of the families of seminorms {a → DaLp(O)}D∈Diff (O) and {a → Op(Da)Sp(H)}D∈Diff (O).

3 Weyl-Pedersen calculus Op: C∞,p(O) → Sp(H) is cont. linear.

Here L0(O) := {a ∈ L∞(O) | lim

ξ→∞ a(ξ) = 0}.

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 10 / 16

slide-11
SLIDE 11

Ingredients of the proof

  • g0 := g/z ❀ G0 acts simply transitively on O
  • Convolution w.r.t. the coadjoint action: b ∈ L1

loc(G0), a ∈ L∞(O) with

compact support, (b ∗ a)(ξ) =

  • G0

b(T)a(Ad∗(−T)ξ)dT for ξ ∈ O. This gives Op(b ∗ a) =

  • G0

b(T)π(T)Op(a)π(−T)dT.

  • Decomposition of the Dirac distribution δ: for all m ∈ N, there exist

finite families {uj}j∈J ∈ U((g0)C) and {fj}j∈J in C m

0 (O) such that for

every a ∈ S′(O) a =

  • j∈J

A∗

ξ0(a ∗ uj) ∗ fj,

where A∗

ξ0(a)(X) = a(Ad∗(−X)ξ0).

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 11 / 16

slide-12
SLIDE 12

Classical applications

  • G = H2n+1 the Heisenberg group
  • Op: S′(R2n) ∼

→ L(S(Rn), S′(Rn)) the classical pseudo-diff. Weyl calculus

  • Not. S0

0,0(R2n) := {a ∈ C ∞(R2n) | (∀α ∈ N2n)

∂αa ∈ L∞(R2n)}

Corollary 1 (Calder´

  • n-Vaillancourt L2-boundedness th.)

a ∈ S0

0,0(R2n) =

⇒ Op(a) ∈ B(L2(Rn)).

Corollary 2 (Beals criterion)

If T : S(Rn) → S′(Rn) cont. lin. and for all r ≥ 1, j = 1, . . . , r, Xj ∈ {Q1, . . . , Qn, P1, . . . , Pn} we have [X1, . . . [Xr, T] · · · ] ∈ B(L2(Rn)), then there is a ∈ S0

0,0(R2n) such that T = Op(a).

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 12 / 16

slide-13
SLIDE 13

Non-classical examples, pag. 1

The main theorem applies to a wide variety of situations, quite different from the Heisenberg groups:

First piece of evidence (L. Corwin, 1983)

Every simply connected nilpotent Lie group embeds into a nilpotent Lie group whose generic coadjoint orbits are flat. ⇒ There exist such groups of arbitrarily high nilpotency index.

Second piece of evidence (D. Burde, 2006)

There exists a nilpotent Lie group with the properties:

◮ the generic coadjoint orbits are flat; ◮ the Lie algebra has only nilpotent derivations.

⇒ Such a group need not be stratified.

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 13 / 16

slide-14
SLIDE 14

Non-classical examples, pag. 2

  • g = span {X0, X1, X2, X3, X4} with the center z = RX0
  • {0} = [g, [g, [g, g]]] = [g, [g, g]]
  • (η1, η2, η3, η4) global (non-canonical) coordinates in the phase space O
  • π: G → B(L2(R2)) unirrep associated with O
  • (q1, q2) coordinates in the configuration space R2

Example 1

◮ [X4, X3] = X1, [X4, X1] = [X3, X2] = X0 ◮ Diff (O) ≃ alg ({∂2, ∂3, ∂4, ∂1 + η1∂3 − η2∂4}) ◮ dπ(X1) = iq1, dπ(X2) = iq2, dπ(X3) = ∂ ∂q2 , dπ(X4) = ∂ ∂q1 − iq1q2

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 14 / 16

slide-15
SLIDE 15

Non-classical examples, pag. 3

Example 2

◮ [X4, X3] = X2, [X4, X2] = X1, [X4, X1] = [X3, X2] = X0 ◮ Diff (O) ≃ alg ({∂3, ∂4, ∂2 + η1∂3 + (−η2 + η2

1

2 )∂4,

∂1 + η1∂2 + η2

1

2 ∂3 + (η3 − η1η2 + η3

1

3 )∂4}) ◮ dπ(X1) = iq1, dπ(X2) = iq2, dπ(X3) = ∂ ∂q2 ,

dπ(X4) =

∂ ∂q1 + q1∂2 − i 1 2q2 2

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 15 / 16

slide-16
SLIDE 16

The role of symplectic Lie groups

  • O ≃ G/Z =: G0

❀ G0 nilpotent Lie group with a left-invar. symplectic structure

  • ω: g0 × g0 → R, ω(X, Y ) = ξ0, [X, Y ]g
  • χ: g0 → g∗

0, χ(X) = (Ad∗ G(X)ξ0)|g0

  • Def. The affine coadjoint action of G0 is

γω : G0 × g∗

0 → g∗ 0,

(X, η) → γ(X)η := Ad∗

G0(X)η + χ(X).

Diff (g∗

0) the algebra of linear γω-invariant diff. oper. on g∗

Key observation

P : O ∼ → g∗

0, ξ → ξ|g0

❀ Diff (O) ≃ Diff (g∗

0)

Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia lowie˙ za, June 27, 2012 16 / 16