Existing answers provide a solid response regarding the idiom in play in Luke. In the interest of supplementing these answers, I'll address the question as it relates to the synoptic problem (which is tagged in the question).
Synoptic Problem
If we grant that this conversation took place in Hebrew (see an argument to this effect in the appendix), then Hebrew idiom is straightforward and Luke, as is typical of his style, preserves the Hebrew structure of the phrase when he writes it in Greek. Mark, as is typical of his style, writes it in more colloquial Greek.
On the Two-Source Hypothesis, Luke copied this account from Mark. On the Farrer Hypothesis, Luke copied this account from Mark, but was influenced by Matthew as well. On the Two-Gospel or the Clementine Hebrew Hypotheses, Mark copied this story from Matthew & Luke. Let us consider both directions of dependence.
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Mark before Luke
If Luke copied this story from Mark, Luke has taken Mark's colloquial Greek and edited it back into a more Hebraic Greek. Luke is writing to a Gentile audience (probably in Greece--see Anti-Marcionite Prologue to Luke). This is a pointlessly complicated editorial exercise.
If Luke's source states in plain Greek "I am", he has no need to incorporate a Hebrew idiom that could potentially confuse his audience. Furthermore, the account in Matthew (see 26:64) also uses Hebrew idiom to convey the message "I am". On the Two-Source Hypothesis, the material shared by Matthew & Luke that is not in Mark came from the hypothetical "Q" source, but Q had no passion narrative. This means that both Matthew & Luke (neither of whom has read the other's work) read Mark's account, and both independently decided to add a somewhat confusing Hebrew idiom to Mark's plain Greek.
This passage is one of what are known as the minor agreements of Matthew & Luke against Mark, which serve as one of the weightier evidences against the Two-Source Hypothesis.
Luke's editorial procedure is unnecessarily complicated on the Farrer & Two-Source hypotheses (editing Hebraic structure back into the Greek of his source), and also wildly unlikely on the Two-Source Hypothesis (Matthew & Luke both independently made this unusual change to the text).
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Luke before Mark
If Luke wrote before Mark, both authors follow their customary writing patterns, and no unusual editing takes place.
Luke tends to preserve much Hebrew structure in his writing, even though he is writing in Greek (see further discussion on this site here), leading some (myself included) to believe that Luke is working with Hebrew source material. If Luke is working with Hebrew source material he's going to run into Hebrew idioms--like this one--and this would be but one of many cases where Luke has chosen to preserve in his translation not just what was said in his Hebrew source, but the way it was said.
Mark later comes across the Hebrew idiom and renders it in "street Greek" (as opposed to formal, literary Greek). Mark uses informal Greek all the time, and this is a straightforward editorial procedure--especially if the origin of Mark's Gospel is oral in nature, which I argue here.
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Conclusion
Luke & Mark are saying the same thing, Luke using a Hebrew idiom, and Mark without the Hebrew idiom. The difference between the texts is best explained if Luke did not use Mark as a source.
Appendix--Hebrew language
As noted by Dottard, Perry Webb, and multiple Biblical commentaries, there is a Hebrew idiom being used here. "You say that I am" or "Thou hast said" is a way of answering the question in the affirmative. In the comments there was discussion as to what language was being spoken at the trial.
My arguments that Jesus would have been able to communicate in Aramaic, Hebrew, and Greek are presented in this video (the "elite" of society in the Sanhedrin would have known these 3 languages as well). The question then is which of these 3 languages was being employed?
Greek is exceptionally unlikely--these are Jews, in Jerusalem, conversing with other Jews from the region, about a matter of Jewish law and Jewish theology. Greek was the language they used to communicate with their pagan overlords & pagan neighbors (and sometimes Jews from other parts of the Empire).
Aramaic is possible...but unnecessary.
Hebrew is the most likely, since their prophetic scriptures (the meaning of which is being debated in this trial) were written in Hebrew. As discussed in this post, the Hebrew scriptures were not written in Aramaic until after AD 70. A Jewish tribunal discussing the Jewish scriptures among locals in Jerusalem is more likely than not to be speaking in Hebrew.
