Applied researcher with deflect and repeat costs

Yes — an Empowered Applied Researchers reduces the total cost of your spell, which includes both the Deflect surcharge and the Repeat cost (Core Rules 356.2 → 356.4; 809.1.d; 820.1.d).

Costs are summed first (base + Deflect's mandatory Power cost + Repeat's additional cost if you choose to pay it), then the discount is applied once to that combined total (same template as Vex, Cheerless vs. Deflect, FAQ #11173, and Eager Apprentice vs. Repeat, FAQ #629/#9301). Example: a [2] Energy spell with Repeat [2] targeting a Deflect [1] unit costs 4 Energy + 1 Power before the discount, then 3 Energy + 0 Power after.

Caveats: the discount only shaves Energy (to its [1] minimum) and Power (any domain), so a Repeat or Deflect cost that's a non-resource cost (e.g., discard a card) isn't reduced; and it applies once per spell — paying Repeat doesn't let you apply the discount a second time. Note this differs from Ezreal, Prodigy, whose discount can't touch Deflect at all because Deflect is a mandatory (not optional) additional cost (FAQ #11207) — Applied Researchers has no such restriction.