Proposition 4. In the event either the Department or the County will exceed its spending limit as determined by Proposition 4, and the voters fail to approve a measure to allow exceeding the limit, this MOU shall be reopened at the request of the County/Department for the purpose of negotiating personnel costs to comply with the Proposition 4 spending limits.
Proposition 4. 15. In the situation of Theorem 4.8, additionally assume that V : D → Set is a faithful and limit-preserving functor. UD , к¸r ,¸ and UÐA = AÐ. See below. A к V r¸ U ,,¸ Ð
Proposition 4. 20. Let Ω be a finite set with the discrete topology. A predicate transformer p : ΩY → ΩX is finitary if and only if p is probabilities). Our goal is a healthiness result in this setting, towards which we rely on our relative algebra recipe. As we noted the original The- orem 4.8 is not enough; we use its finitary variant (Theorem 4.19), providing its ingredient (a relative algebra Ω) by means of the lifting result (Proposition 4.12). D It turns out that the category in the relative algebra recipe is given by so-called generalized effect modules. They have been used in the context of categorical quantum logics [19] and the more general theory of effectuses [5]. continuous with respect to the product topology of ΩY and ΩX . Definition 5.3 ( ). A partial commutative monoid (PCM) GEMod ❽ ❽❽ ❽ ❽ ❽ ❽ ❽ is a set M with a partial binary sum and a zero element 0 ∈ M that ' ' '
Proposition 4. 19. Let G be a compact connected Lie group and T a maximal torus. Write δ : T → C for the Weyl denominator function. Then there exists c > 0 such that f '→ c|δ| · f|T defines a unitary map L2(G)AdG → L2(T)W (G,T). Proof. See, e.g., [12, Corollary 3.14.2]. In Theorem 3.15, we showed that the Dolbeault–Dirac quantization of T∗G with its standard K¨xxxxx structure is HL2(T∗G, e−|Y | ε), which is G × G-equivariantly isomorphic to L2(G) via the isomorphism of Proposition A.1. This isomorphism composed with the Weyl integration formula produces a canonical isomorphism between the reduction after quantization and L2(T)W (G,T). On the other hand, in Theorem 4.17 we applied the isomorphism u in Proposition 4.16, and again Proposition A.1 to also identify QDD(T∗G//AdG) with L2(T)W (G,T). Hence: Theorem 4.20. Defining the Dolbeault–Dirac quantization of T∗G//AdG to be the Dolbeault–Dirac quantization of its principal stratum, (Dolbeault–Dirac) quantiza- tion after reduction and reduction after quantization are both canonically isomorphic to L2(T)W (G,T), and hence quantization commutes with reduction. Quantization of the cotangent bundle of a compact Lie group
Proposition 4. 5. (1) If H ⊃ T, then j−1(0) ∩ (G × g)(g,Y ) = (G × g)(g,Y ) ⊂ T × t.
Proposition 4. 8. Let H ≥ T be an isotropy group for the adjoint action of G on G × g. Then Γ∞c (T × t\(G × g)H, E) is dense in Γc∞(T × t, E) with respect to the graph norm of D, the Dolbeault–Dirac operator on T × t. Proof. Choose ϵ > 0 such that the exponential map exp : g → G is a diffeomor- phism from the open G-invariant neighborhood U = {Y ∈ g | |Y | < ϵ} onto an open neighborhood V ⊂ G of eG. Let H ≥ T be an isotropy group for the action of G on G × g. For each we consider the chart (g, Y ) ∈ (G × g)H = GH × gH, (exp−1 ◦Lg−1) × id : gV × g → U × g. Because g is fixed under H, this diffeomorphism intertwines the H-action. Conse- quently, exp−1 ◦Lg−1 maps (gV × g) ∩ (G × g)H onto UH × gH ⊂ t × t ⊂ g × g. i=1 Choose an orthonormal basis {ei}t of t such that {e1, . . . , et−k} spans gH ⊂ t, where k ≥ 1. By re-ordering the coordinates induced by the orthonormal basis {e1, . . . , et}, the subset (G × g)H ∩ ((gV ∩ T) × t) is mapped onto OH := {(x1, . . . , x2t) ∈ (U ∩ t) × t | x2t−2k+1 = · · · = x2t = 0} ⊂ t × t under the coordinate chart (exp−1 ◦Lg−1) × id : (gV ∩ T) × t → (U ∩ t) × t for T×t. Under this chart the K¨xxxxx structure on T×t corresponds to the standard K¨xxxxx structure on O := (U ∩ t) × t ⊂ R2t, because the differential of exp : t → T is trivial when the tangent spaces on T are identified with t through left-trivialization. Because GH is compact, one can choose finitely many elements gl ∈ GH such that {(glV ∩ T) × t} covers (G × g)H = GH × gH . Then U := {(glV ∩ T) × t} ∪ {T × t\(G × g)H} is a finite open cover of T × t. Consider the chart gl (exp−1 ◦L −1 ) × id : (glV ∩ T) × t → O Quantization of the cotangent bundle of a compact Lie group ˜ ˜ ˜ for some fixed l. Let f ∈ Γc∞(O, E). When O is regarded as an open subset of R2t, f can be extended (by zero) to a section in Γc∞(R2t, E). Recall that the chart (exp−1 ◦Lg−1 ) × id maps the K¨xxxxx structure on T × t to the standard K¨xxxxx structure on O = (U ∩ t) × t. Therefore, the operator D on Γc∞(O, E) is just the restriction of the ordinary Dolbeault–Dirac operator D on R2t to O. Note that for compactly supported sections on O the graph norm with respect to D is the same as the graph norm with respect to D, as the operator D is local. ˜ ˜ By Corollary 4.2, a section s ∈ Γc∞(O, E) can be approximated in the graph norm of D by a sequence (sm)m ∈ Γc∞(R2t\R2t−2k, E). Let ψ : R2t → [0, 1] be a smooth function with compact support contained in O such that ψ ≡ 1 on supp s. By Lemma 4.9 below, ψsm → s in the grap...
Proposition 4. .2. If δ = 0 and γ − r > σ2 , then there exists y∗ > q such that g(y∗) = 0 and the y∗ is unique. So b = ay∗ > q is unique too, where g(y) is defined by (3.19). σ2 2 Proof. Since δ = 0 and γ − r > σ2 , we have λ = 2(γ−r) > 1 = λ . It is easy to prove g˜′′ (y) ≥ 0. By an argument similar to the proof of Proposition
Proposition 4. .1. If δ > 0 and γ − r + δ ≥ 0, then there exists y∗ > q such that g(y∗) = 0 and the y∗ is unique. b = ay∗ > q is unique too, where h(y) = λ1+1−λ2 y1−λ2 − q λ1−1−λ2 y−λ2 , g(y) is defined by (3.19). λ1 a λ1−1 Proof. Since δ > 0, we have λ1 > 1 > λ2, q q λ2+1 q λ1−λ2 and g( a ) = ( a ) (1 − ( a ) ) < 0 ∞ lim g(y) = . y→∞ a By continuity of g(y), there exists y∗ > q such that g(y∗) = 0 and b = ay∗ > q. Moreover, it is easy to see from the procedure in section 3 that the assumptions in Theorem 3.1 hold for the b. Next we prove the uniqueness of y∗. Define g˜(y) = y−λ2 g(y) = (λ − 1)yλ1 +1−λ2 − q λ yλ1 −λ2 + (1 − λ )y q Then
Proposition 4. .1.2. For λ, µ ∈ X+(n), Homt(∆n(λ), ∇n(µ)) = 0 unless λ = µ in which case it is 1-dimensional. We can now state the following result from [13].
Proposition 4. 1.4. For λ, µ ∈ X+(n) the space HomtL(n)(∆n(µ), ∇n(λ) ⊗ 2 Vn∗) is zero unless µ = λ−ϵi − ϵj for some 1 ≤ i < j ≤ n such that j is λ-removable and i is λ−ϵj-removable. V Proof. Recall from Section 2.4.2 that the elements of Wn acting on the weight (0, . . . , 0, −1, −1) permute its entries. In particular the orbits of (0, . . . , 0, −1, −1) under Wn is {−ϵi−ϵj|1 ≤
